How to enable on screen keyboard at startup menu?

Look for a salvation army or other thrift store,

you should be able to pick up a USB keyboard for 3-5 $$$

When you start the machine, are you brought straight to the login window and asked for a password? Or do you have your account set to bypass the login window and log right in. If the former, I am not of aware of a method log in, or bypass the log in, without a keyboard.

Then in that case, I don't know of a solution. If you were to bypass login, you could enable on screen type assisting (type w/ your mouse, essentially). But bypassing login requires a keyboard, AFAIK. Best of luck.

If you had a second Mac and had previously enabled ScreenSharing on the Mac mini you could connect them and use the second Mac to control the Mac mini and effectively be the keyboard.

If you have even a PC around you might be able to use its keyboard if it is USB or if there is a computer store (Mac or PC) you could buy a cheap USB keyboard. If your budget stretched further and there is an Apple store around you could buy an Apple Bluetooth keyboard which should auto-pair to the Mac.

As necessary, this giveg mapping for using a PC KB

Keyboard mappings using a PC keyboard on a Macintosh