Just take this as a learning experience (once you have fixed the mess this client has made, which to a degree, you let her make. But you can fix this!).
The big, waving red flag was when she refused to sign a contract but you took her as a client. None of your clients should be without a contract and payment up front; that is how most child care operates and you should not make any exceptions.
So first: You tell her today both by e-mail and by phone -- so she doesn't turn up with her kid tomorrow -- that if she does not sign the contract AND pay what is already owed as well as a certain amount (two weeks? A month? What do you do with other clients?) in advance at the same time, you and she are no longer doing business. Don't let her wheedle or plead you into "Please please just take my child this one day tomorrow, or just this one week...." Tell her you have done a firm and final review of your business practices and are instituting this policy and be clear it's not just for her but for all your clients. ALL of them.
Then make it so. Give current clients a short time, maybe two weeks, before you fully implement this with good notification to them but after that: Only written contracts, signed before any child walks through your door. Statements on late pickup fees that are set in stone and specify that traffic problems, dead phones, flat tires will still mean they incur the late fee, no exceptions. Does the current contract include statements of your fee structure and due dates of payment (first of the month sounds typical for most types of business, or last of the month)? A firm fee on overdue payments and termination of your services if a payment is X days or weeks overdue? Include in the contract your rules regarding food provision for children (do parents bring it, do you provide, etc.) and issues like diapers (for each child, the parents must provide you with X number diapers minimum per day etc. if that would work for you....) And do this for all clients including ones you already have. Some may bail on you if you are now letting them come late or put off payment but tell them you are firming up all your contracting and payments because this is a business. (If you already have all that in the contract you now use, great, and please forgive me, I'm just brainstorming!)
If the particular client in question here says, as you noted in your post, that she "doesn't agree with" something in your contract, then you invite her to find a different day care provider. Again, don't let her plead her way into stayiing on with you.
I'm assuming you are trained and/or state-licensed--? Have you sought out any online forums for professional day care providers where they can give you better advice than we can here? There may be some form of association or professional group online where child care providers can trade tips and advice and that would be excellent for you to seek out when these kinds of things occur!