I think there is quite a lot you can do yourselve at home without necessarily having the expense of finding a tutor.
I posted some advice on the Warwickshire forum recently which is switching to Durham CEM 11+ (the same organisation that sets the KE gerammar tests). I think the advice is just as relevant for KE grammar 11+ exams - understanding vocabularly is absolutely key for these exams:-
Univ of Durham CEM Bespoke 11+ tests have been used for the Birmingham KE Foundation Grammar exams for a few years now. No doubt the new new warwickshire test will have some similarities.
The skills the candidates will need to solve the cloze test sound similar to those required for VR synonyms sector of KE Birmingham tests; viz. extensive vocabularly, reading age circa 14.5 years + coupled with an understanding of the true meaning of words.
In the synonyms tests I was given an examples by prep schools teachers such as:-
Volatile, load, explosive, talkative, welcoming... where you have to select a word from the last 4 that best matches the 1st word.
To give you an idea of the expected level, other words might typically be....Vibrant, Indolent, Labyrinthine, Enigmatic, Abhor, Zealot, Wrangle, Brevity, Notorious etc etc.
This is definitely an area you can work on with your child. You can do a number of exercises:-
1. Get your child to read some articles of quality broadsheet newspapers every day and get them to explain what's going on.
2. Read through the newspapers yourself and select and highlight some of more difficult words with a highligh pen and make sure they understand the precise definition.
3. Maybe try 11+ material such as the The Tutors Volcabularly Builder- see 11+ material section (Please note I haven't tried this myself - but it sounds like it might help)
4. Just do lots of different VR, NVR and Maths material.
5. Keep them topped up over the school holidays
Hope this helps
Hope this helps