No idea on a FireHD as that is in fact an Android tablet not a kindle of the e-reader type. You can log in on BL and redownload the files but I have no idea how the FireHD handles mobi or epub files.
Maybe look into OTG cables (On the Go) cables, which let you plug in an HD to a tablet, that might be a generally good diea as a stopgap till you get a new real computer.
That said, the dead PC, was it a desktop or laptop? If a desktop, it is quite possible you can fix it by swapping parts out.
Wait! There may be a way, you can set up an email address you send files to to get them onto a kindle. If you donwload the file to the tablet, and you can attach it to an email to the address, that might work.