Looks like your lawn is Brown patch crazy and I agree with turf md that you most likely are causing the rapid decline of the turf by overwatering and creating the environment perfect for this disease to thrive.
The over watering is not only obvious by the dollar weed present in the lawn but also explains the huge outbreak of brown patch.
I know you say things are tight right now and you can not afford professional service, but think about this, how tight are things going to be when you have to resod that lawn and the HOA is on your tail to replace? You need someone who can treat the brown patch as well as help you ensure your irrigation system is getting good coverage so you are not overwatering to compensate for poor design. As a general rule you should be watering 1 time a week now until daylight savings time change. Then you can figure on twice a week. As far as time nobody is going to give you a perfect answer over the internet without knowing the layout of heads, the output of the system and area to be covered. A general rule that we tell customers is to run pop ups for 25 to 30 minutes, rotars for 45 minutes to a hour once a week during the winter and twice a week at those times during the summer. This usually gets them in the area where they can fine tune the irrigation from there however this by no means is perfect as there is lots of things to consider such as is the pop up in a full shade area or full sun. Your lawn appears from the pictures that it can be saved vs resodding at this point but if conditions are allowed to stay the current courses you will be resodding very very soon.