Keyboard backlight is working properly, but keyboard itself - doesn't

I assume you are using an external keyboard & mouse right now.

Go into the keyboard prefs and enable Show Keyboard, Emoji & Symbol Viewers in menu bar. Open the keyboard viewer and see if any of the keys are working as you press the keys the viewer will show you the key hit. Do any keys work? Is there a pattern on what is working Vs not working?

If the battery had become swollen then you might only need to replace the ribbon cable from the logic board to the track pad as the pressure may have damaged it. If on the other hand the keyboard & trackpad had something spilled into them you'll need to think about replacing them which is a bigger fix.

Follow this IFIXIT guide: MacBook Pro 15" Unibody Early 2011 Trackpad Replacement it will walk you though getting to the ribbon cable and trackpad if its bad too.

If there was something spilled into the system then you'll need to follow this IFIXIT guide: MacBook Pro 15" Unibody Early 2011 Upper Case Replacement. You have two choices on to do the repair, Replace the full upper case which is more expensive but is a lot quicker or carefully remove the keyboard subassembly from the upper case replacing the keyboard but it's a lot of work as you have a ton of screws and/or pop rivets to deal with. You should also make a detailed inspection of your logic board for any corrosion damage as well.