What are my rights if my house has been infested with bird mites but the exterminator has done nothing and now our things are ruined?

First, in terms of the landlord's liability: it depends upon whether, on the circumstances, the landlord took "reasonable"--which is not the same thing as effective or optimal--steps to address the infestation. You write that it was infested for over a month and the exterminator came 2 weeks ago. It is not clear that is an unreasonable delay. Say that you told the landlord about the infestation 4 - 5 weeks ago, and the exterminator came about 2 weeks after that. Since it is not unreasonable for the landlord to wait a day or two before calling for an extirminator (extirmation is not like a fire or water-gushing leak; it may be urgent, but it's not an emergency in the same way), then given the exterminitor's schedule and weekend days, getting him there in 2 weeks may be slower than desirable, but would not be considered by most landlord-tenant judges to be unreasonable.

Similarly, the exterminator has apparently not gotten rid of the infestation yet, but many infestations cannot be or are not cleared up after one treatment--multiple treatments is not uncommon. If the exterminitor will be coming back for follow-ups, again, that is not unreasonable.

The reason this is important is that the landlord's obligation is to do what he reasonably can; he is not held to a standard of perfection. Based on what you write, it is not at all a given that you have a viable claim yet against the landlord, though you may if there is no effective follow-up.

If and when you do have a claim, you may be able to break your lease if the problem is continuing and is not or cannot be resolved, because the landlord will be violating the implied warranty of habitability. You are unlikely to recover the cost of your belongings if the "damage" (them being ruined) was done in the first few days of the infestation, since no landlord could have prevented damage that quickly, and landlords are liable for not doing the impossible. But if the damage occured due to what are later found to be unreasonable delays, then you might be able to recover compensation.

Generally you cannot get moving expenses--what you get, if appropriate (i.e. a continuing, intractable problem) is out of the lease and the abiltiy to move.

Since the specific facts (when you reported this to the landlord; when he called for the inspection; is there follow-up by the inspector; etc.) are so critical, you are advised to speak in detail with a landlord-tenant attorney about the matter, to see what your options are.