PS/2 to XT Wireless Keyboard Project

So I already have a project working that converts PS/2 and AT keyboard signals to XT.

I would like to expand it to make the keyboard wireless.

I was thinking of using a 433mhz RF receiver and transmitter to do this.

My question..

I can use the power on the computer to power Arduino and the receiver with no issue. to power the arduino and keyboard for the transmitter side I can use the standard AA or even a 9v batteries but I'm not sure how long they would last or I can go with a lithuim ion rechargeable battery with charger board.

Thoughts, has anyone done anything like this before that can be me some guidance and how long could say a arduin pro micro be powered before having to change the standard batteries?

What is XT?