You should immediately contact a local SD personal injury attorney. You didn't mention if it was a public or private school - it might make a very important difference because most states have a very short statute of limitations for filing against school districts - if the statute passes no claim can later be made. The attorney will have to investigate this claim to determine if there was adequate supervision of the playground; adequate inspection of the swing itself or a regular basis; what caused the chain to break; how long it had been in that condition etc etc. I have handled school yard cases & they are often difficult. I am a California attorney, not licensed in SD & not familiar with their laws & statutes of limitations. Call an attorney ASAP!!!
This is not intended to be legal advise or as legal representation. I am a California personal injury attorney . Be aware that every state has its own statute of limitations; and statutes & case laws that govern the handling of these matters.
I agree you should contact an attorney immediately considering the circumstances. Evidence of the condition of the chain will be very important it seems - try to get pictures of it before it is fixed?
As already mentioned, whether it is a public or private school will make the case very different - suing the city gets a little more complicated.
Speak to someone in your area who is familiar with personal injury and suing the public school district. As the other attorneys have mentioned, notice is an important requirement, and differs from State to State.