I'd recommend asking more widely for recommendations of a tutor. Talk to people whose tutoring needs and objectives were the same as yours. Word of wisdom: unless you are asking your sister, best friend of many years or someone from an entirely different year group (with whom your child poses no 'threat' of competition), be very wary of believing the 'I don't know of any tutors/ we don't have a tutor' response. I've had parents in my hallway having a 'stand off' when one said to the other: "I thought you said you didn't have a tutor!" Generally, parents do not share details until they've 'finished' with their tutor. It's a secretive and sadly selfish attitude born from fear of failure. Awful!
Although it's best to find a tutor who specialises in meeting the objectives you have in mind, your child's class teacher may be able to help by asking colleagues or 'teacher friends'. When you DO find someone who has a suitable tutoring specialism (eg 11+ state super selectives for your area, dyslexia, etc.), expect for there to be a waiting list and for it to be £££ if in or near London - which is why it may be worth finding an interim solution.
Alternatively, if you're one of the fortunate parents whose child will listen to you as you give feedback and input, follow instructions without much resistance and you find that you're able to explain concepts that they don't understand in areas of the curriculum related to the particular examinations you're hoping they'll pass, GO FOR IT! Have a look at elevenplusexams.co.uk for guidance.
Much of the work is effectively done at home by the pupil. This isn't to say that working under the guidance of a really good tutor isn't invaluable - it's so helpful for you to feel reassured and guided as to what to work on and for your child to have a positive and experienced tutor to work on areas that are a mystery to them.
Be suspicious of tutors offering to 'assess' your child (particularly if this is at a higher rate) or of those quoting 'success rates'.
Assessment: I've tutored children who've simply not had effective class teaching input, resulting in many gaps in their knowledge and understanding BUT who were/ are so incredibly bright that they've galloped ahead at a rate of knots with the right tuition. Equally, I've taught children who were already working close to their potential (having been better taught at school) so presented with fewer 'gaps' but whose 'rate of absorption' was slower. Another factor, inevitably, is whether practice is carried out at home. Visiting a tutor once a week inevitably accrues very little compared to those consolidating what's been learned, between lessons.
Success rates: those tutors are inevitably fibbing or selecting only those pupils who'll maintain their own statistics - not working hard to get EVERY child to their potential.
I hope that's helpful.
Good luck!