[H] extending range

[Winterlight]

I have a Belkin wireless  #1Router F5D7231 working in a home that has 
very, very poor reception outside of line of site,... with any 
wireless remote device. The #1 Bilken works OK in two thirds of the 
house. The end third of the house= MBR has no reception whatsoever. 
So I got a second identical #2 Belkin F5D7231 on Ebay and ran a CAT 
5e into the room on that side of the house which I plugged into a LAN 
port on the  #1 router thirty feet away.

The routers have a wireless bridge mode that does not work because 
there is no reception of the #1 router to the #2 router in the MBR. 
So, on the #2 router, I enabled it as a WAP and assigned it a IP 
number  on the #1 routers LAN.

Can I now set up the #2 WAP with the same SSD and WPA-2 pass phrase 
as the #1 router is using so that I get a seamless connection  as you 
move from one part of the house to another?  Will this work? Or how 
should I go about this?

thanks
Winterlight