こぼして汚してしまってごめんなさいって英語でなんて言うの？

Spillage and everyday accidents can leave us feel a lttle awkward...but rest assured its all in a day's work for someone in the catering trade!
It is sufficient to point out that you realise a mess has been made ....and that its you (or your family's fault) .

When you want to apologize for your child who spilled something, then you can say:
-I'd like to apologize. My child spilled this.
-Please accept my apologies for spilling (soup on your cloth).

*I apologize on behalf of my child for the mess they have created.
On behalf means you doing something for someone. In this case since your child cannot apology you are apologizing for them
The mess they created means the mess that have caused.
What is important when you are apologizing is that you are sincere about it.

Yes, the neatest way to deal with this situation is to apologise profusely:
"I'm really sorry about the mess!"
To apologise profusely: "Profuse" means "in great excess." If you say about someone, "Oh, yeah, he apologized profusely," you're saying, "he apologized way more than he needed to." Someone who apologizes profusely has apologized more than once, and they've essentially made more of a big deal over their transgression than they should....however it shows they are really sorry!
Or, you could just say: "So sorry about that!" These things happen all the time in restaurants where there may be children. It's no big deal!

"I'm sorry for the mess." is used as a way to apologize for something being made dirty because of you or someone with you.

The two sentences you see above are excellent ways to apologize to a restaurant employee for the mess that a child made. In the first sentence you will see the term make a mess. That means to make something unclean or dirty. This is a term we use a lot in our everyday conversation and would make a great addition to your vocabulary.