All right, have you tried giving your DS Lite the IP I suggested previously?

Also if I recall correctly, you can't access a DS/Lite's wi-fi settings unless you have a game with on-line capabilities and head to the wi-fi settings from there. This might sound a bit stupid, but I think it wouldn't hurt to use a different game. Just in case.

Anyway the last thing I can think of is this:

If you're using a Windows computer, go to Start and type in 'cmd'. (If you're in an older version of Windows, go to 'Run...' and type in 'cmd')
Type in exactly the following: ipconfig /ALL
A huge pile of information should pop up onto your screen. I'll get to that in a bit.
Go back to your DS Lite and attempt to configure it automatically once more.
Back to your computer screen, look for the term "IPV4 Address". The number to the right represents the IP of your computer. Now add 10 to the last bit of your IP. For example, if you had:

IPV4 Address. . . . . . . . 192.168.1.102 (example)
Then type 192.168.1.112 into your DS Lite.

You can also try to take different bits of information (Default Gateway, DNS, etc) and copy them straight to your DS Lite (obviously with no modifications).
This always works like a charm for me when I can't seem to configure it automatically.

If it still doesn't work then I would have suggested you get yourself a new Wi-Fi chipset for your DS Lite, but what bothers me is that it either works or it doesn't. It's strange that it can detect your access point but not connect to it when your DS Phat can. You might want to just try getting the chipset from the link provided. It's only $1.66 anyway and it ships worldwide for free. Oh and while you're at it get yourself a tri-wing and a Philips screwdriver, since you can't open up your DS Lite without them. This is, of course, assuming you're comfortable with opening up your system. If you decide to go that route, I can give you precise steps. I've disassembled and re-assembled DS Lites countless times.