Having a child who is considered highly gifted (my son is now in his teens) I understand what you mean about boredom. I'm not sure where you live, but I'm in a larger city, and I found a great deal of information about things to do by visiting the library. Not only did the library offer some special programs for children (such as skits and magic shows and arts and crafts) but they had postings and newspapers that announced various activities for children in the local area (like nature walks and museum hours) and of course, there were always rows and rows of books to look through. I would often let my son select books he liked and then we would use that as a topic for our "theme of the week." (For example, if he grabbed books about animals, we would read the books, and then perhaps spend an afternoon visiting a local pet store, etc. If he grabbed a book about weather, we would try to measure temperatures and create charts.) Cooking together, of course, is a wonderful way for children to learn about science and math, as they measure different ingredients, etc.

I did introduce my son to piano lessons at your daughter's age. He really disliked having to practice, but he did seem to benefit from the several years he took the lessons. Be sure whatever instructor you choose is familiar working with young children. You might ask the music teacher at your child's school or the music/choir director at your church if they know of anyone who gives private lessons. Let us know how it goes!

Yes, she's repeating kindergarten due to her age. She's already done kindergarten in the UK last year and was nearly a year ahead then in Maths. She can read and write. I believe the synesthesia is just with numbers.
I could change her to a full time class here, paying a fee, but since she's already completed the work that seems like an expensive childcare exercise.
I want to be able to stimulate her mentally rather than specifically academically or for the synesthesia. She seems bored and unfocused a lot of the time. Her brain's working overtime but she doesn't know what to do with it! I think in due course she will be identified as gifted and talented (I expect every parent says that!). I've signed her up for a few classes at the YMCA these are mostly physical activities and what she really needs is mental stimulation. I'm not experienced at dealing with this and don't know where to start! I wondered about trying her with a musical instrument?

Hi. Is your child being forced to repeat kindergarten in the U.S. now, due to some kind of age limitation? Did you have an option to have her attend an all-day program, and if she's bored in a part-time program, can she change to the longer class now? When you say that you want to stimulate her, is that to say that you want to further promote her general academic growth, or her synesthesia? I'm not personally familiar with the condition, but I understand In one common form of synesthesia, letters or numbers are perceived as inherently colored. You already mentioned numbers, but does your daughter have a similar ability with her letters? Is she reading already? I realize I'm asking more questions than I've answered, but if you help us to better understand what you're seeking, we can try to give you a better response...