From Mar. 14 til Apr. 6 is 23 days. You should have a mated queen by then, either laying or getting ready to. If they started a queen cell the day you split, she will likely have emerged by now, so don't expect to find queen cells when you inspect it. You most likely have a virgin queen. I would not order a queen before Apr. 12.
A frame of brood and eggs never hurts a hive, so if you can obtain one, it will just be good insurance.