If you paid someone to do it (any internal repair on a 2nd gen shuffle), it would probably cost more than the $49 to get a new 4th gen shuffle.
If you are handy with fixing electronics, you can buy a replacement battery (with tools) for under $10 on eBay. There are online guides on doing the repair, for example
http://www.ifixit.com/Device/iPod_Shuffle_2nd_Generation
I'm not sure it's the battery, though... I have a 2nd gen shuffle with a dead (or nearly dead) battery. It works as expected while connected (it's recognized by iTunes), but the battey does not hold any charge, so as soon as I disconnect it from power, it is completely dead. (I've been meaning to try the repair myself; just have not gotten around to it.)
So your shuffle's problen could be something else, like the logic board.