What is it that's stopping you from sending a contract and then invoicing your client? When you say that you need confidence, I'm wondering where in your thought processes did you decide that you don't deserve to be paid for your work? I understand that there has been no discussion about rates or how many hours the client wanted, etc. so now you have fifteen hours in and you know your client has no frame of reference for the invoice that you need to send. Fair enough!
I agree that what you *need* is an agreement - and you need it set up and signed before you complete even an hour of work for any client. Where I disagree with Shelley is the implication that the client should have mentioned a contract, etc. - you're a business owner/service provider - you are the only one responsible for putting your systems in place and binding clients to them. But that part is really neither here nor there...
Prep and send her a contract today with information about your rates, policies, etc. Don't offer reasons as to why it's late, just send it and ask her to sign it. Let her know that her invoice will be coming at x date. In order to preserve the appearance of regular business operations, I'd suggest setting that date at one month since the time you started working with her. As in: regular invoicing happens once a month.
And then I'd create a system around all of this so that you have your contracts and invoicing ready to go for future clients.
I really don't mean to sound harsh at all, if I do... but if you're going to find long term business success you have to start thinking like a business owner
Please do post back and let us know how it all works out! Best case, your client will sign and smile and pay her invoice when the time comes