Englischkurs is correct.
If I read an offer for an
Englisch Kurs I would think I get lessons from a native English speaker, because the title seems to be defined by somebody who speaks English - not by somebody who speaks German.
Englischer Kurs sounds grammatically correct, but it does not mean language lesson. You may use it, if a school offers lessons in English and French. Then you have an englischer Kurs and a französischer Kurs. Or maybe you have a cooking lesson in English and French. Then you may say englischer Kurs to explain which lesson you mean.
Sprachschule is correct, Sprach Schule is not correct.