BTW, the new information is that the other person could not work out the answer.
1st day
A sees 8 towns. A knows that B sees 2 or 5 towns, but does not know which.
A knows that
if B sees 2 towns, B will know that A sees 8 or 11 towns
if B sees 5 towns, B will know that A sees 5 or 8 towns.
A knows that A doesn't know how many towns there are, and that B doesn't know how many towns there are. However, B does not know that A does not know how many towns there are. If B sees 2 towns, B will know that A sees 8 or 11 towns, and if A sees 11 towns, A would know there were 13 towns.
A knows that A doesn't know how many towns there are. But there's a possibility that B knows that there's a possibility that A knows.
1st evening: Neither knows, they stay imprisoned.
2nd day
A now knows that B knows that A does not see 11 towns (because if A saw 11 towns, they would both now be free). A knows that if B sees 2 towns, B will now know that A sees 8 towns. But A also knows that it's still a possibility that B sees 5 towns.
A knows that A still doesn't know how many towns there are. But A also knows that there's a possibility that B now knows how many towns there are — if B sees 2 towns, he knows that A sees 8.
2nd evening: A doesn't know. As it happens, B doesn't see 2 towns, he sees 5, so he still doesn't know whether A sees 8 or 5 towns. They stay imprisoned.
3rd day:
A now knows that B does not see 2 towns, as if B saw 2 towns, B would have known on the second day that A saw 8 towns not 11 towns, and they'd both now be free.
So A now knows that B sees 5 towns.
3rd evening: A knows that B sees 5 towns, so tells the Gaoler that there are 13 towns, and A & B are freed.
HTH