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I'm so sorry you lost your chicken.
The tan coloured bugs are lice and they will not kill her but are usually an indication that a bird has been sick for a while and not looking after herself by preening and dust bathing. They are more of an irritant than anything else, whereas mites suck their blood, which obviously creates more serious health issues. Mites live and breed in the crevices of the coop and crawl onto the chickens at night to feed.
Chickens will hide illness until they are too weak to do so, hence you not noticing anything amiss sooner.
How old are your birds and how many do you have with the two roosters? Do they have access to a dust bathing area as that will help them to keep these pests at a manageable level themselves. To treat an outbreak, people usually use a permethrin based spray or powder. I appreciate that a previous poster has suggested garlic juice and I would be interested in the scientific evidence to back that up. Not saying it won't work but I am wary of claims of anything working 100%.
My feeling is that there is some ailment or disease that has caused the death. Getting a necropsy done or even conducting an informal one yourself can be surprisingly informative. Some of us take photos of out post mortem exams and share them and discuss possible cause of death. It is surprising how often you can pinpoint a serious issue even without medical training. I accept it is not something everyone can get their head around though.
I wish you well with the rest of your flock.