How to login when I can't enter my password because "e" key on keyboard doesn't work

Try entering ⌘+option+F5 to bring up the "Accessibility Options" popup window. This window will appear similar to what is shown below.

Select "Enable Accessibility Keyboard", then click on the "Done" button. The on-screen keyboard should appear, as shown below.

Use the keyboard to enter the "E" key or the entire password.

Apple's current Bluetooth keyboard is the Magic Keyboard, and this also allows the keyboard to be connected to your Mac via a supplied Lightning adapter cable.

I would suggest connecting this first in order to be able to pair the keyboard, and then you will be able to log in as usual using the external Bluetooth keyboard afterwards.

You only need any physical keyboard (PC, Mac, anything basically) to set up your Bluetooth and for emergency use if Bluetooth pairing fails in the future.