I claim to be no expert I would not try to charge the mouse and cell battery at the same time. However this setup has worked great for me for recharging these old batteries. I have 1 of these batteries powering my keyboard for the last 30 days and only lost 1 bar of power, I got about 18 hours of power for the brass horse lamp before recharging. And keep in mind that these batteries are not new.
FOR SAFETY REASONS I SUGGEST YOU DON'T ATTEMPT TO DO THIS! UNLESS YOU KNOW AND UNDERSTAND BATTERIES AND BATTERY TYPES ALONG WITH CHARGERS AS I AM BY FAR NO EXPERT AND KNOW JUST ABOUT ENOUGH TO BE DANGEROUS.
Step 1: Tools & Parts
Solder & Flux
3.7 Volt Cell phone battery
USB Cable with male & female ends