Where I work they usually use a larangeal mask like Kim said (also known as a fanny on a stick) the fanny part fits over the laranx and then air via a syringe is inserted into it to form a nice soft seal to keep the airways open. The stick part is where the oxygen/gas goes down. Usually they use a type of gas in with air or oxygen to keep the kidlet sedated if its a short procedure like grommits so no IV sedation is needed. Where I work most have a cannula for IV fluids although for grommits that probably isn't neccessary depending on how long your DD has been NBM. They sedate them with the gas and if an IVC needs to be inserted it is done when they are under GA as it is not traumatic when they are asleep. My daughter had grommits also (but was 3) and she was in and out in recovery within 15minutes. It was very quick.
DD was not sleepy at all after her grommits. Working in recovery we are usually looking for a parent very quickly as waking up in a strange place with strange nurses often sends them off as soon as they wake up. I am sure it will be very quick for you and your DD.