In most conditions, 肯 is similar as 愿意. For example: Are you willing to lend me a pencil? (你愿意/肯借把铅笔给我吗). The difference is 愿意 is written, 肯 is colloquial.The meaning of 高兴 is just happy.

1.Firstly, "高兴" is a adjective, "肯" and "愿意" are auxiliary verbs. 2. "肯" has more strong meaning than "愿意" when express one's thought. But most of the time, they can replace by each other. 3.Examples. 她不肯让我玩这个娃娃。（ I have tried many ways to persuade her ,still failed at last.） 她不愿意让我玩这个娃娃。( She in unwilling to play with this doll, maybe I would try another way……)

This three words can make different results in different circumstance

I think 肯 is more conversational than 愿意. For example, if you are writing an email to your boss, you would use 愿意 instead of 肯 to express the same meaning. What do you mean by difference of 高兴, any context?

你高兴和我一起打篮球么？你肯和我一起打篮球么？ 中国人不那么说． 你愿意和我一起打篮球么？It's right !