|If you are getting the XP install screen then it sounds as if XP is not installed. In that case you can't recover the product key. I'm afraid that you will need to purchase a new copy of XP or, better still, take the chance to upgrade to a newer OS. If money is a problem you could always try one of the free OSs like Linux.|
In any case, you don't need to purchase a new computer, just new software. Be sure, in future, to keep a careful note of Product Keys, Serial Numbers and the like. That is, in essence, what you are paying for (especially true now that much software is purchased as a digital download).