Keyboards keys are delicate! I would focus on swapping out the damaged parts to the good upper case system (broken screen) or just replace the [product|IF108-055-3] part. Basically, transferring the parts to the good upper case. Here's the guide [guide|16998]
Given the age of your system I'm sure your battery is not very healthy if its still the original!
Let’s get a better view of things, install this gem of an app! [https://www.coconut-flavour.com/coconutbattery/|CoconutBattery] take a snapshot of the apps main window and post it here for us to see [guide|21499]
Here's the needed battery if you know its time [product|IF108-063-2] and the guide [guide|16840]