Does your router have multiple ethernet ports? Is your 10 Mbs LAN a repeater or switch?
Most routers these days have at least a 100 Mbs multi-port switch built in.
A different approach would be to backup all the files on a portable device like a USB terabyte storage and then load the data onto the other system.
The router is SUPPOSED to be 100mbs switch. I cut out the router by running a Cat 5 to the other computer directly with a X over box. Started copying again and it was locked at about 10mbs. For me that makes trouble in the boxes. The HD’s are both capable of over 100mbs and I can copy files within one computer at 50+Mbs to a second drive. Even if I take drive C from one and put it in computer 2, my network speeds will still suck when I go back to normal. Putting the drive in the other computer will speed up this file copy job, but the network will remain snail slow.
I have USB 2,0 speeds on both boxes. That's slower than what I'm doing now as I would have to copy twice to get the job done.