I'm not sure if this is in violation of the "Do not just post links or quotations" rule here since it is from /r/linguistics and not AskHistorians, but there is an answer listed in the faq there that specifically mention Munich and Cologne.

Here's part of the answer.

In the case of Munich, it's first mentioned in 1158 as "forum apud Munichen," meaning "place by the monks." "Munichen" is the Old High German dative plural of "munih," meaning "monk." In modern German, that's now "Mönch." In English, we've dropped the last syllable, essentially the case and number marker (English has a bit of a history of doing that with case markers). In German, they've dropped the second syllable and umlauted the accented syllable.

So it's not as if an English speaker went to Munich, asked them what they called the place, and when presented with "München," said, "The tits with that, this place is Munich." Instead, both English speakers and German speakers over the course of centuries gradually changed a word, much like any other word in the language.

In another case, Cologne, Germany (Köln in German), comes from the Latin Colonia Claudia Ara Agrippinensium. Here, the English seems somewhat more conservative than the German.

I don't have the knowledge to explain exactly how each current English pronunciation was arrived at, so I won't attempt to answer that. However, I would strongly disagree with your premise that those German names are easily pronounced in English. Both contain sounds which simply do not exist in (most varieties of) English. The standard pronunciations of the German names are given below, rendered into IPA.

/ˈmʏnçən/

/kœln/

Most varieties of English have no such phoneme as /ʏ/, /ç/, or /œ/.

Because Munich and Cologne are the French names for München and Köln respectively, and the English names for those cities were taken from French instead of German. The same applies to places like Basle (in German, Basel), Lucerne (Luzern), Berne (Bern) and, archaically, Mayence (Mainz).