I assume you paid by credit card. Since LH argues it was not you who flew. You can give them a choice either they accept and credit the miles or you will challenge the credit card payment. Since you did not fly, you did not book and than why should you pay.
This is just an argument not a legal position to stand up to.

I once had lady on the LH check in who got crazy because my girlfriends surname was spelled uncorrected. After a few minutes I had to tell her very straight forward to f**** off.
Than a supervisor noticed the argument an suddenly: no problem at all. The lady than tried to turn it. As if she just wanted to protect us, because there might be problems with immigration in SA. But believe they do not care even if they would realize that the name is spelled incorrect. Believe it or not the miles were posted straight away.