Broad mites are resistant to Neem.
If you want to besure you must eliminate or strongly cut back the plant or use a mite specific acarizide.
I use a product that contains abamectin, it shows an effect within a few days and has a waiting time of only 3 days for peppers.
Organic growers use collodial sulphur but this has no curative effect.
As mites are distributed by the wind and by insects, it doesn´t make much sense only to treat the plant that shows signs of damage.
All other plants around may carry the mites as well.
Peter