Are you referring to one of those small black jumpers? There not made to be used a lot. They have a small lithium batteries that aren't made to handle the current needed to jump a car too many times. I use these kind of batteries in my RC trucks. If too much current is drawn the batteries will puff and risk catching on fire. Just my 2 cents
Maybe I'm missing something, but it sounds like your battery has just reached the end of its' life. Seeing how after a week sitting on a charger and just barely enough umph to start your car...sound like a kaput battery to me.
old post, but you can usually replace the lead acid battery for 20-30 dollars in your defunct unit. Take it apart and find out what size is used, and find a replacment on ebay.