Well, this all sounds pretty horrible. Do you mean to say you don't actually have a floppy drive on the laptop? I'm not sure about the integrity problems with the DVD, it sounds more like the hardware of the CD/DVD player is failing. As for tftpd32, I don't think this will work if you can't login to Windows anymore...
The DVD installer might also be failing because it runs out of memory (512 MiB is the minimum requirement, so it should be fine, but still...). If you have a separate swap partition for Helena, figure out what the device name of that is (/dev/sdaX, where X is a number, probably). You can run "sudo parted -l" to figure it out, or start GParted from the menu. Then before
you start the installer, activate that swap partition also for Linux Mint 11 with command "sudo swapon /dev/sdaX" (where /dev/sdaX again matches what you found earlier).
Another suggestion might be to burn the ISO of Plop
to a CD, and then boot from that on the failing laptop (it is very small, so should load fast hopefully without problems). Then from Plop continue the boot to Linux Mint 11 which you have put on a USB stick with UNetbootin or something similar.