V.C.
I would definitely decline any future flu shots for your daughter, report this to your doctor, ask if they will report this reaction to VAERS (Vaccine Adverse Event Reporting System), and look into the possibility of egg allergy or allergy to any other component in the shot. Has she had noticeable reactions to other shots? Did she receive the thimerisol-free form of the flu shot (Flusone - single dose syringe)? Did she receive any other shots that day? I would delay and separate her upcoming 12-15 month shots (MMR, Varicella, etc). With my daughter, I waited to 18 months for the MMR, am doing the DTap and Hib at 21 months, and will do the varicella at 2 years. I am not getting flu shots for either of my kids this year (and their pediatrician did nothing to try to convince me otherwise...hmmm). Only hubby got one, so if we get sick and he doesn't, we'll find out if the flu shot is actually effective this year. Incidentally, a Cochrane Review of all research studies related to pediatric flu shots showed there is no evidence that the flu shot works for children under 2.
There is a 12-hour stomach bug going around that causes vomiting, headache and general low-energy malaise (my son had it last week), but your mention of the sudden runny nose makes me think allergic reaction.
I hope she is better soon.