Every once and awhile, I have non-Swedish guests that inquire about going to a theatre to see a play. They seem genuinely surprised when I explain that plays and musicals are generally in Swedish (with the exception of operas) and not English. This is, of course, because the vast majority of theatre goers in Stockholm are Swedes and it is also not possible to fill a theatre with non Swedes every evening for the typical run of a show.
However... every now and then a play is put on in English. A good example of this is the Maxim Theatre which puts on a production of Charles Dickens' "A Christmas Carol" every December. But you don't have to wait until December to see a play in English here in Stockholm. The Stockholm English Speaking Theatre Company (SEST) regularly puts on short run productions in smaller theatres. Next up is "The God of Carnage" by Yasmina Reza which will run between the 15th and 25th of February at Teater Brunnsgatan Fyra. Otherwise keep your eyes on their website (linked above) for future productions... like perhaps a Shakespeare in the park this summer?