Hi Eli,
I am replying to this post as I do specialise in exactly that area - fun English games for children aged 4-12.
As far as photocopiable resources these can be life savers. However I also believe that teachers often photocopy far more than they need, and with some imagination, a few pictures and a class board one simply does not need to do hundreds of photocopies of activity sheets, which take the children 2 minutes to fill in.
Have you tried playing some class games (not necessarily pairwork, but purposeful games)? That might help keep all levels of students interested in the class at the same time.
I highly suggest looking into games to play with the students you tutor. At first thought, you might think that it can be hard playing a game with just one student there are a number of things that you can do, such give a limited time for the task using a stopwatch or timer. The student must complete the task before the timer goes off so you add excitement by having a race against the clock. You can also time your student and then see if they can beat their time in the next round.
You can also play too and again there are things you can do to make it a genuine game, rather than one where you either win all the time or pretend to be slow. For example you can give your pupil a head start or make your task harder.
You can find out more about teaching children with games one on one right here and watch a demo lesson;
Teaching a child English
I wish you all the best with your tutoring. It's highly rewarding and a valuable skill because you can use the same games and techniques to teach your own children or family members a second language.