Im just wondering whether you could use Ultron's second power when he is already exhausted.

"If Ultron would become stunned, instead you may discard a card. If you do, lose 8 endurance and exhaust Ultron."

I think that you would be allowed to because exhausting him isnt part of the cost, but my friend thought I was wrong, so whats the right answer?

Its not part of the cost, nor does it require Ultron to be ready in order to use it. Therefore, even if Ultron is exhausted you can still use the ability.

It's just there to make sure Ultron doesn't get to attack affter the replacement effect saved him.