My return and menu buttons are constantly illuminated and not working.

I installed a replacement LED screen. Now the return and menu buttons are constantly on and not working. Everything else works fine. I tried setting the "touch key light duration" to off and they are still illuminated. I removed the battery and rebooted, Samsung Galaxy S4 logo comes on for a moment and the return and menu buttons turn on, the colored Samsung and stars shows for a moment and the ATT logo on a white screen comes on, the "swipe screen to unlock" notice comes on, after a few moments the screen goes black. While all this goes on the return and menu buttons remain illuminated even when the screen goes black. Please help! Also, if I plug in the charging cable the green "battery charging" logo shows and the return and menu buttons illuminate. The red led in the upper left is also on when charging.

How confident are you that it was not a faulty screen?

Welcome to Android Central! The capacitative buttons aren't actually part of the LED screen, right? You probably disrupted something during the screen replacement process. At this point, you may want to take it to a qualified repair shop to see what went wrong.