If my apartment complex has not provided me with hot water for a week and now my child is getting sick, can I sue them for unsanitary conditions?

Assuming you have provided the complex with notice that you do not have hot water, then you may be able to sue them; however, the amount you can recover may be limited. When a rental unit has something impairing its habitability--like a lack of hot water--tenants may be entitled to a rent abatement, or pro rata rent reduction, for  the time they live with the impaired condition. This means that they get back some percentage of their rent for a period of time. For example, since people can live without hot water, say that lack of hot water would entitle you to a 60% reduction in rent for the time you lack hot water--it would almost certainly not entitle you toa  100% reduction. Say your rent is $1,200 per month, or approximately $300 per week. You would, in this example, potentially recover up to 60% of $300 for each week you've been  without hot water, or $180.

Note that the above is just by of example: a court could find you entitled to a greater or lesser percentage abatement. More significantly, though, a court could also order the landlord to take action to cure or correct the problem, if the landlord is unwilling to do so him/herself. You should speak wiith a landlord-tenant attorney about your situation; one option, if you can't afford a lawyer, is to contact Legal Services, since they provide legal assistance to people who cannot afford an attorney. Good luck.