Sorry, I'm confused-The ultimate aim is to reinstall ME or to upgrade to XP?.
If you want XP, remember you can't install XP twice, legally. When you register it for the old machine, MS will void your license for the new one. If you have Auto Update off on the new machine, that might make no practical difference to you, but your new machine would then be running an illegal OS.
Edit to add - and if you don't register it, XP will shut down in 30 days, which as Tami says may be no bad thing.
If your sole aim is to reinstall ME on the old machine, then you need to know the answer to this question-
Is the ME disc a full install, an upgrade or a manufacturer- supplied "recovery " disc? If the last, there should be a custom reinstall program already on the machine, which will have codes in it that should match your ME disc. The original documentation of the comp should explain how to use this. The program may be on a hidden partition, which, I hope, you have not somehow erased.
If your ME was an upgrade, you might be best to try uninstalling ME and reverting to 98, or whatever you had before, then reinstalling onto that. This can have problems. Be certain you have the CD and floppy drivers for the earlier system on a bootable floppy as you may have to install them before you can reload the earlier OS.
Nb- if the earlier OS was itself an upgrade, I seriously advise you to bite the bullet and buy another XP upgrade disc. It's just not worth the hassle.
If your ME disc is a full install version, revert to what ZEP said.
Back up everything on your new comp, reformat and start over.
I assume there is no hardware problem like a bad C drive.