In which parts of Greece do people pronounce the word “και” as “che” instead of “ke”?

Lots. Your search term is tsitakismos, the Greek name for the affrication of palatal /k/ [c] to [tʃ, tɕ, ts], as exemplified by the pronunciation of /ke/ “and” as /tʃe/ instead of Standard Greek [ce].

Going through the Centre for the Greek Language’s writeup of Modern dialects, and looking for that tsitakismos keyword:

• South-Eastern
• Some islands in the Dodecanese
• Chios
• Cypriot (as [dʒe])
• Northern Dialects
• Lesbos, Skyros, Mykonos
• Cappadocian (refugee, moribund)
• Southern Italian Greek (not in Greece)
• At least some parts of the Peloponnese
• Tsakonian
• Old Athenian (including Aegina, Kyme, Megara)
• Cretan and the Cyclades

In fact, as I’ve already mentioned in another answer, somewhere, the better questions to ask is: Where does tsitakismos not happen? The answer appears to be the Greek mainland and the Ionian islands, some bits of the Peloponnese and Attica excepted.

Incidentally, Greek dialectologists seem to have only worked out in 1983 (Contossopoulos’ paper) that the real split between Greek dialects is the word for “what”: τι vs ίντα, separating Aegean Islands and Old Athenian from the Mainland. That split corresponds closely to the tsitakismos isοgloss. The three Northern dialect exceptions, for example, are islands.

EDIT: cc Dimitris Almyrantis Philip Newton I’ve just stumbled on this paper from last year by Pantelidis: http://ins.web.auth.gr/images/ME… , which fascinatingly suggests that there are old connections between Old Athenian and Peloponnesian, and concludes that Old Athenian used to be spoken in Eastern Central Greece, which now speaks Northern Greek or Arvanitika. He notes, pp. 307–8, that tsitakismos turns up in bits (bits) of both the Peloponnese and Central Greece, though there is just too little data to work out how old and extensive the phenomenon is.