Turn a AA power supply into rechargeable battery pack

[Pen16]

So I am completely new to soldering and designing my own electronics but I thought this might be a good place to start.

I have a logitech K350 Wireless Keyboard that runs on AA batteries. I use it all the time in the living room for family internet and movie watching. The mouse it comes with works long enough on the rechargeable batteries that its not a problem however the keyboard eats the energy.

I have been saving my old cell phone batteries and my old PS3 controller batteries and I was wondering if there is a way I can solder some new components in and make it a rechargeable keyboard with maybe a micro-usb re charger.

I wanted to come to a community of people that were not only good but supportive as I have no real previous experience other than making a good solder. I would appreciate any help in the right direction and thank you in advance.