Legally, there's nothing wrong with giving her a check. The problem comes in when she doesn't cash them. If she doesn't cash them, then you have no proof you paid then either. You're back to square one!
Instead of depositing cash into her account... write a check and put THAT in there. Put on the memo line "Child support for (month)" and put that puppy right on in her account just like you do the cash. There will be your cancelled check you can get a copy of. You'll have your "proof" you want (a check) and she'll have her money in the bank on the same day she's used to it being there.
You're cutting off your nose to spite your face here and really making this harder than it needs to be.
But, if you're mailing her checks and she's just trashing them, guess what? She's going to say you haven't paid, and what are you going to have as proof you did? Certainly not a cancelled check since she never cashed it and is playing games.
Don't mail a check to her. Put one made out to her directly into her account. It's the same thing you've been doing, only you are depositing a check instead of cash and you WILL have a paper trail this way. And the best part? She can't "refuse" it, throw it away, or just hang on to it to "get you" and say you're in arrears and haven't been paying because she will never have it in her own hands!
As an aside, I've been banking since I was 16 years old. Over 20 years. I have never seen a deposit slip from a bank or credit union that didn't have the amount of the deposit on it. Even an ATM machine receipt gives you the deposit amount and the ending balance.
Edit to add: If you're going to ask how you can deposit it into her account without her signing it....
On the back of the check where the signature line is, print in BIG BOLD LETTERS:
FOR DEPOSIT ONLY