Англійський is a masculine adjective.
Here we can see an example of nominalization: instead of англійська мова (English language) we commonly say just англійська (українська, німецька,...) - just like in English. After the nominalization the adjective keeps the gender of the noun it used to be an attribute of: мова - feminine.
"По-англійськи", "по-українськи" means more "in english (or maybe England) manner", "in Ukrainian manner". Like in "Котлета по-київськи" - I suppose some of you heard of that. The meaning would be: Like it's done in Kyiv.
We also use a phrase "Піти по-англійськи", wich means "To leave without saying goodbye". But I don't know where did it come from :).
"Говорити по-англійськ-и/-ому" is probably also acceptable, but I wouldn't say that. I would say "Говорити англійською" or "Розмовляти англійською" - it is much better.