I personally would not try repairing the old faucet. You can pick up a pretty inexpensive new, shiny faucet for just a few bucks more than the full repair kit, which is needed to do the job properly.
Yes, they are very simple to install. Just make sure your valves are able to shut off the water completely before digging in....and just follow the simple instructions. Since Valentine's Day is right around the corner, consider this as a gift to your Mom! She will love a nice, bright faucet!
If you need help dealing with the new install, don't hesitate with asking one of us here!