Validação de login

<%@ page import="java.util.*" %>
<%@ taglib uri="http://java.sun.com/jsp/jstl/core" prefix="c"%>

<jsp:useBean id="validaAdmin" class="dao.admin.TestaLoginAdmin" />

<c:set var="admin" value="${validaAdmin.testaLogin(pageContext.request)}" />
<c:set var="redireciona" value="${validaAdmin.redireciona(pageContext.response)}" />

$

<c:choose>
  <c:when test="${admin != null}">          
            Olá ${admin.nome}
          <c:set var="tipo" value="${admin.tipo}" />
          <c:choose>
            <c:when test="${tipo=='s'}">          
              <style>#super{display:block;}</style>
              <style>#comum{display:none;}</style>
            </c:when>
          </c:choose>
          <c:choose>
            <c:when test="${tipo=='c'}">          
              <style>#super{display:none;}</style>
              <style>#comum{display:bloq;}</style>
            </c:when>
          </c:choose>
  </c:when>
  <c:otherwise>
             $
    </c:otherwise>
</c:choose>

<%@ page import="model.Admin"%>
<%
HttpSession sessao = request.getSession();

Admin admin = (Admin) sessao.getAttribute("admin");
String tipo = admin.getTipo();
out.println("Olá "+admin.getNome());
sessao.setAttribute("tipoAdmin", tipo);
	  
if (sessao.getAttribute("nomeAdmin") != null)
{  
    sessao.removeAttribute("nomeAdmin"); 
}
	  
if (tipo.equals("s"))
{
    out.print("<style>#super{display:block;}</style>");
	out.print("<style>#comum{display:none;}</style>");
} 
else if (tipo.equals("c")) 
{
    out.print("<style>#super{display:none;}</style>");
    out.print("<style>#comum{display:block;}</style>");
}
%>

<%@ page import="java.util.*" %>
<%@ taglib uri="http://java.sun.com/jsp/jstl/core" prefix="c"%>

<jsp:useBean id="validaAdmin" class="dao.admin.TestaLoginAdmin" />

<c:set var="admin" value="${validaAdmin.testaLogin(pageContext.request)}" />
<c:set var="redireciona" value="${validaAdmin.redireciona(pageContext.response)}" />

$ ===

<c:choose>
  <c:when test="${admin != null}">          
            Olá ${admin.nome}
          <c:set var="tipo" value="${admin.tipo}" />
          <c:choose>
            <c:when test="${tipo=='s'}">          
              <style>#super{display:block;}</style>
              <style>#comum{display:none;}</style>
            </c:when>
          </c:choose>
          <c:choose>
            <c:when test="${tipo=='c'}">          
              <style>#super{display:none;}</style>
              <style>#comum{display:bloq;}</style>
            </c:when>
          </c:choose>
  </c:when>
</c:choose>

<c:choose>
  <c:when test="${admin == null}">
             $
  </c:when>
</c:choose>

E recomendo você utilizar mudar a implementação do seu método redireciona, pois no processo de renderizar a página o método redirecina referenciado é executado e isso já faz sua app ir para a pagina de index e isso impede a exibição do nome do admin

package dao.admin;

import model.Admin;

import java.io.*;  
import javax.servlet.*;  
import javax.servlet.http.*;

public class TestaLoginAdmin 
{ 
	public Admin testaLogin(HttpServletRequest request) throws ServletException, IOException 
	{
		System.out.println(request+" = Request");
		HttpSession sessao = request.getSession();
		Admin admin = null;

		admin = (Admin) sessao.getAttribute("admin");

		if(admin != null)
		{
			return admin;
		}
		return admin;
	}
}

<%@ page import="java.util.*" %>
<%@ taglib uri="http://java.sun.com/jsp/jstl/core" prefix="c"%>

<jsp:useBean id="validaAdmin" class="dao.admin.TestaLoginAdmin" />

<c:set var="admin" value="${validaAdmin.testaLogin(pageContext.request)}" />

<c:choose>
  <c:when test="${admin != null}">          
            Olá ${admin.nome}
          <c:set var="tipo" value="${admin.tipo}" />
          <c:choose>
            <c:when test="${tipo=='s'}">          
              <style>#super{display:block;}</style>
              <style>#comum{display:none;}</style>
            </c:when>
          </c:choose>
          <c:choose>
            <c:when test="${tipo=='c'}">          
              <style>#super{display:none;}</style>
              <style>#comum{display:bloq;}</style>
            </c:when>
          </c:choose>
  </c:when>
</c:choose>

<c:choose>
  <c:when test="${admin == null}">
             out.println("<script>location.href="index.jsp";</script>")
  </c:when>
</c:choose>

<jsp:useBean id="validaAdmin" class="dao.admin.TestaLoginAdmin" />

<c:choose>
  <c:when test="${admin != null}">          
            Olá ${admin.nome}
          <c:set var="tipo" value="${admin.tipo}" />
          <c:choose>
            <c:when test="${tipo=='s'}">          
              <style>#super{display:block;}</style>
              <style>#comum{display:none;}</style>
            </c:when>
          </c:choose>
          <c:choose>
            <c:when test="${tipo=='c'}">          
              <style>#super{display:none;}</style>
              <style>#comum{display:bloq;}</style>
            </c:when>
          </c:choose>
  </c:when>
  <c:otherwise>
             $
    </c:otherwise>
</c:choose>

  <jsp:useBean id="validaAdmin" class="dao.admin.TestaLoginAdmin" />
  <c:set var="admin" value="${validaAdmin.testaLogin(pageContext.request)}" />
<c:set var="redireciona" value="${validaAdmin.redireciona(pageContext.response)}" />

public class Redirecionador {
    private HttpServletResponse response;
    private String pagina;

    public Redirecionador(HttpServletResponse response, String pagina) {
        this.response = response;
        this.pagina = pagina;
    }
    
    public void redirecione() throws IOException{
        response.sendRedirect(pagina);
    }
}

 
  public Redirecionador redireciona(HttpServletResponse response) throws ServletException, IOException {        
        return new Redirecionador(response, "index.jsp");
    }

<jsp:useBean id="validaAdmin" class="dao.admin.TestaLoginAdmin" />
<c:set var="admin" value="${validaAdmin.testaLogin(pageContext.request)}" />
<c:set var="redirecidor" value="${validaAdmin.redireciona(pageContext.response)}" />
<c:choose>
  <c:when test="${admin != null}">          
            Olá ${admin.nome}
          <c:set var="tipo" value="${admin.tipo}" />
          <c:choose>
            <c:when test="${tipo=='s'}">          
              <style>#super{display:block;}</style>
              <style>#comum{display:none;}</style>
            </c:when>
          </c:choose>
          <c:choose>
            <c:when test="${tipo=='c'}">          
              <style>#super{display:none;}</style>
              <style>#comum{display:bloq;}</style>
            </c:when>
          </c:choose>
  </c:when>
  <c:otherwise>
             ${redirecidor.redirecione}
    </c:otherwise>
</c:choose>

package filtro;

import java.io.IOException;
import java.util.ArrayList;
import java.util.List;

import javax.servlet.Filter;
import javax.servlet.FilterChain;
import javax.servlet.FilterConfig;
import javax.servlet.ServletException;
import javax.servlet.ServletRequest;
import javax.servlet.ServletResponse;
import javax.servlet.annotation.WebFilter;
import javax.servlet.http.HttpServletRequest;
import javax.servlet.http.HttpServletResponse;
import javax.servlet.http.HttpSession;

@WebFilter("/admin/*")
public class AdminFilter implements Filter 
{
    private static List<String> allowedURIs;
    
    public AdminFilter() 
    {
        // TODO Auto-generated constructor stub
    }

	public void destroy() 
	{
		// TODO Auto-generated method stub
	}

	public void doFilter(ServletRequest request, ServletResponse response, FilterChain chain) throws IOException, ServletException 
    {         
	       HttpServletRequest req = (HttpServletRequest) request;  
	       HttpServletResponse res = (HttpServletResponse) response;  
	       HttpSession sessao= req.getSession();  
	       
           if(
        		   sessao.getAttribute("admin")!= null || 
        		   precisaDireitoAcessoURI(req.getRequestURI()) || 
        		   req.getRequestURI().equals("/imwsaogotardo/admin/") || 
        		   req.getRequestURI().equals("/imwsaogotardo/admin")
        	 )
           {
               chain.doFilter( request, response );
           }
           else
           {    
               res.sendRedirect("index.jsp");  
           }    	       
    }

	public void init(FilterConfig fConfig) throws ServletException 
    {
        if(allowedURIs == null)
        {
            allowedURIs = new ArrayList<String>();
            allowedURIs.add("/imwsaogotardo/admin/index.jsp");
            allowedURIs.add("/imwsaogotardo/admin/retornos/");
        }
    }
	
	public boolean precisaDireitoAcessoURI(String uriAtual) 
	{

	    for (String uri: allowedURIs) 
	    {
	        if (uriAtual.contains(uri))
	            return true;
	    }
	 
	    return false;
	}	
}