If your iPad is in recovery mode, i would suggest you try to update only first with a new battery. As you said, Battery is not holding charge so firstly you need to use a new battery to take ipad out of recovery mode by updating only if it works otherwise restore is the final step. you can use itune or other software available some time helpful if itune does not work. after that you can test your old battery and ipad if it is charging well.

It seems like related to Audio IC issue which is a very common problem with iPhone 7 and 7P. Some broken connection under the Audio IC is responsible for this issue. Audio IC has a pad c12 which is connected to a near by resistor R1103, you need to repair that broken path by putting a jumper wire between C12 & R1103. This is the right solution for this problem. Some time in rare case may need to repair J12 and H12 pads also.

you need to check the battery health and condition first. Type or paste powercfg /batteryreport /output "C:\battery-report.html" into the window and press Enter to run the command. Open Windows File Explorer and access the C drive. There you should find the battery life report saved as an HTML file. Double-click the file to open it in your preferred web browser. If battery health is good then you need to go to repair shop.

Try resetting the phone after taking the full backup. if still problem persist, you need to check any liquid damage on the board or on the button cables.