How are you cooling down a penis to fit it in the cage?

For example if I want to put it in the morning I usually do have an erection.
So how can I quickly eliminate it without cumming?

Ice works not really well because even a little touch to penis restores the erection. or am I doing something wrong?

Yes, taking it off is your mistake. Leave it on. Give the key to your wife or girlfriend and your problem will solve itself.

I need to wash it sometimes

How about distraction! Do something unerotic like bathing or showering, watching TV, etc. Then put it on.

All these, like watching TV, take time.
I usually wear cage in the morning and my wife wants to ensure that everything is ok.
She is not interested in waiting so we are looking for options as short as it possible.

A simple solution is a swift beating with a cane, the stinging pain is a good temporary distraction!

I know it takes time, but the "doing something non-erotic" suggestion is a good one. But it doesn't have to be time-consuming if you do the things you would normally do during that time. You mentioned this was in the morning - so how about go make yourself (and your wife, if she gets up at the same time) some breakfast. Read the paper (or a news or other non-erotic website) while you eat. Or otherwise, do whatever your normal morning routine would be. By the time you are done, you'll be soft. Then you spend 5 minutes at the end putting on the cage before leaving for the day.

Pull your balls way out and just power it on. It's a rush.