How to bill a client when budget was never discussed

Bill them your regular rate and add an amount for a rush-job and mark it clearly in your invoice. Be open to explain or even negotiate, as other users have already suggested. In your question there is a lack of communication, so you can make a first move towards more and detailed communication with a well written and fully detailed invoice. Once they realize that you note your hours and costs, they will less likely “drop” work on you like that. And do not forget that you accepted the job and also did not mention money, so tread gently.

I know such situations as a customer and as a provider:

I am working for a non-profit organisation but that does not mean that we do sloppy planning and last-minute hectic jobs. When we are late with a project, we cannot expect an external specialist to “catch up” for us. They need to learn to respect their partners even more than fully-paying customers.

What I like doing, when people are bringing work to us (not graphical, but typesetting for example) is telling them our normal rates per page.

And when they underestimate the amount of work and what time it takes to do a good quality job and they want it “tomorrow”, then I tell them it normally takes several working days… “But you can have it by tomorrow, if you pay a bonus for a rush-job. Our team will gladly work all night for you, if you pay zzz (and then I name a very high rate, like four-fold price).”

We are non-profit, so we do not want to charge a lot. But we want to train our partners to never come last-minute and to respect our office-work flow. Same as we respect those people who do printing for us or other work by order, not excluding craftsmen.