The logic of connection is similar to FireWire and SCSI. So you shouldn't have any problem using the spare Thunderbolt port on the display with a longer Thunderbolt cable.
If you went to an Apple Store they would even test it out on their gear to be doubly sure before you bought the cable.
With that said it's possible the damaged Thunderbolt display cable could cause problems, so I would replace it just to be sure. This is a very easy fix.
Thunderbolt-MagSafe combo cable: Apple Assembly # 922-9941
Here's one I found on eBay [http://www.ebay.com/itm/USED-922-9941-All-in-one-cable-Apple-Thunderbolt-Cinema-Display-27-inch-A1407-/200965217408?pt=LH_DefaultDomain_0&hash=item2eca75dc80|USED 922-9941 All-in-one cable]