How do I get my mail app to let me sign back in

What steps have you tried? Have you tried uninstalling and reinstalling again? Have you gone to the settings app and tried from there?

Have have you held the power button and home button to do a soft reset?(for about 6 seconds)

You can try to reset network settings under general,reset. You can try to reset all settings if that doesn’t work. DONT HIT ERASE AND RESET. Make sure you select reset network or reset all settings.

let me know how it goes if you need more help.