Hi four chickens. I really sympathise with you. I've been in a very similar situation this year. I've not managed to go more than 3 weeks this year without taking a day off because one of my children has been ill. Add to that nursery shutting because of building maintenance and my childminder having a poorly child too and I have pretty close to going insane several times this year, including this month.
I know that I'm very, very lucky indeed, as although I have no family, I have a couple of really good friends who have helped out, and I also have absolutely wonderful bosses who have never once given me a hard time about the amount of time I've had off. I am able to work from home to some degree though, which has helped, but I understand completely where you are coming from about the constant feelings of guilt - whatever you do, you feel like you're letting someone down, whether that's your poorly child or your employers/colleagues. I don't have any words of wisdom about how to deal with it, but just wanted to let you know someone understands.
On a more practical note, see what you can do about your DS's health. If he's getting recurrent tonsillitis, then surely he's a candidate for a tonsillectomy? My DD had hers out 2 months ago and the difference is already very, very obvious. She is a different child and SO much healthier. Also, if you haven't already, get your DS referred to asthma clinic at the hospital. Even with colds his asthma should be better under control than it sounds (unless the poor mite has the brittle variety?).
I know your situation sucks. I am sincerely hoping 2010 is better for me, and now for you as well. But hang on in there and know you're not alone. If we get through this at least we'll be able to feel justifiably proud of what we've managed to cope with.