Annoyingling is not a real word...
Did you mean "annoyingly"? If you did, it means: in an annoying way. It's an adverb.
Or, if somebody coined that word according to the word roots, it'll mean:
annoying: causing annoyance; irritating; bothersome.
-ling: one having (such) a characteristic.
annoyingling: one having such a irritating characteristic.
If I didn't help you, I'm so sorry, but this is the best I can do.