I attached my solution I think it is not correct but you might be able to see what I did and tell me how to correct it
I don't think it works. First, you did not properly introduce a and b: what exactly are they? More seriously, you started with hk = kh and arrived at h2k2 = k2h2. You assumed what you are supposed to prove.
The idea is to note that hk = kh ⇔ hkh-1k-1 = e ⇔ (hkh-1k-1 ∈ H and hkh-1k-1 ∈ K). This has been described several times in this thread.