As you suggest and
If using an agency, ask for the same tutor plus full info on him or her
Qualifications
What ages and stages already being taught/have been taught by tutor
Track record of results - documented if at all possible (references from former pupils perhaps)
Where will the lessons take place - if not in your home, see the location before taking the tutor on (is it quiet, coducive to study, adequate space etc)
Individual tutoring or with others?
How will progress a) be measured and b) communicated to you and your DD?
Policy for cancellation by either side and arrangements for catching up
What if there is no improvement? What will be a reasonable period in which to judge that?
How will the tutor expect you to reinforce learning? How will that be communicated to the tutor?
How will you react if the tutor tells you that DD has little chance of success? This is one for you to think about.
I tutored children with whose parents I was always honest. If they were borderline I said so. Some parents would not accept that their children were not up to standard and that putting them through the exam would be traumatic. Are you communicating to your DD that not getting through the exam will be, effectively, the end of the world? Or, are you communicating that the philosophy is for her to try her hardest but if it doesn't come off you are proud of the extra hard work that has been done. Will you be gutted if she does not get a place at the school and is that/will that be communicated to DD?
I did not have contracts (but this was some years ago when attitudes to education were generally less pressured). Advice on that will doubtless be given by other posters.