In reply to Baku Steve:
Many times I am amazed at the things people on here find to be paranoid about, but in this case you're spot on and absolutely right to be very concerned indeed. Rope can melt extremely easily in this kind of situation so your choice to lower incredibly slowly, combined with the thickness of the rope loop, doubtless saved you from a broken back or worse.
If ever you're in the same situation again, with the same knowledge, you would be well advised to leave a karabiner behind rather than take the risk again.
However, a little extra knowledge can go a long way. Abseiling is a basic skill and easy to learn. If you don't have a belay/abseil device that can take double ropes, use a large screwgate and Italian Hitch.
Another suggestion would be to source a supply of good quality Maillon Rapides (screw-up chain links) and to keep one on your harness at all times for situations like this.
Glad you got away with it this time.