Bird mite advice please!!!

Does anyone have any experience of bird mites?

Robins nested in a broken vent leading into our utility room. 

4 out of 5 chicks have died, most in the last 24 hours we believe due to magpies as one of the chicks was found stuck in the grate; as if something tried to pull it out of the nest. My husband discoveres the deas chicks this morning and has cleared the 4 dead out of the nest as it was starting to smell.

Anyway, there is one survivor who has been hopping around on the lawn today and as there are no chicks left in the nest the kites are starting to migrate...into our house.

I have taped a plastic bag over the vent inside the house and we plan to remove the nest and mites as soon as the nest is not in use and repair the grate.

We have a couple of bird boxes on order from the rspb to provide alternative accommodation in the future because I know they like to return to the nesting site. 

My question is, are the mites going to get out of control?! A quick Google talks about fumigation!! I'm hoping by taping up the vent and hopefully blocking their access and getting rid of as many I can I can hold then at bay until we can clear out the nest.