I have a DIR-825 and DAP-1353, which might be probably similar. can you access the DAP's address in your browser? it is usually something like http://192.168.0.xxx. if you can get to that page, then try following the wireless setup wizard. i prefer that over the push button approach.
During setup, give the DAP a different ssid and use as different channel than your DIR. I gave the DAP a static IP address (inside the DAP setup) and then reserved this address in the DIR's DHCP section. I don't know if that matters though. I used wpa security on both.