Do you know of any types of insurance that a private landlord can purchase, which includes legal assistance coverage?

Unfortunately, I cannot answer that question. What insurance products are available on the market — and which companies might offer legal assistance coverage to private landlords in your situation — is not a question about the Danish Rent Act, it's a question about the insurance market. I don't have updated or reliable information about specific insurance companies' products in my context, so I'd rather be honest than guess a name or coverage that might not exist or be outdated.

What I would recommend: contact an insurance broker or call a few insurance companies directly and specifically ask about legal assistance coverage for landlords (not just tenants) in private rental situations. It might also be worth asking a private landlords' association if they have experience with which companies actually cover the landlord side.

However, if you end up in a specific dispute with your tenant, I can certainly help you understand what the Danish Rent Act says and check your correspondence before you send anything.