Link showing opening T1 to replace screen.
https://www.youtube.com/watch?v=LqbYWHJk8Tk
I also found link to download service manual. It tells how to replace battery and has battery part number. Could not find replacement battery on line. Service manual gives tip on soldering battery cable to circuit board. I would cut each battery wire close to old battery and solder new battery wires to old wires which are still attached to circuit board. This way less chance to damage circuit board by soldering battery wires directly to circuit board.
Shield circuit board from dripping solder.
https://www.ifixit.com/Device/Sony_Reader_Wi-Fi_PRS-T1
I have not found replacement battery on line that is listed as correct for T1.
Easy to find one that would power the reader but they are unknown size and most likely wouldn't fit in reader.