|Are you sure it's the PSU that's clicking. Failure to boot is usually related to the HDD which can make a clicking sound.|
eMachines are famous for PSU failures which , quite often, cause other failures. HDDs Motherboards...etc.
You may be able to recover the Files from the eMachine HDD by use of an Adapter or external HDD enclosure that connects to your LT via USB cable. Lots of options available.
If it's the HDD that's clicking you may have problems recovering Data.
Are you sure, I mean SURE, the 2nd PSU was compatible, or even working?
There is nothing to learn from someone who already agrees with you.