How is Friedrich Nietzsche's last name pronounced, with a long e sound as in "cheese", or with a short a sound as in "ha ha"?

Depends where you live. "Friedrich Wilhelm Nietzsche (German pronunciation: [ˈfʁiːdʁɪç ˈvɪlhɛlm ˈniːtsʃə]; in English UK: /ˈniːtʃə/, US: /ˈniːtʃi/[1]) (October 15, 1844 – August 25, 1900) was a 19th-century German philosopher and classical philologist."

If you can't read those symbols, they say that the UK pronunciation is knee-che (the second syllable containing the ambiguous vowel wound somewhere between a short a and a short e); the US pronunciation is knee-chee.

https://en.wikipedia.org/wiki/Friedrich_Nietzsche

That second syllable is what is known to singers as a "schwa". Depending on the language, schwas may or may not be considered actual syllables. In Hebrew they never are, in German they usually are. In any case, a pronounced schwa is generally pronounced as a short e without stress, almost as if the speaker is just releasing the enunciation of the consonant pronounced just before the schwa. (The closest pronounciation in English is an I as in "kin", but shorter.) Whatever the case, it is never a long vowel, nor should it be sung over two notes.