何度も and 何度か: repeated emphasis and an unspecified few times
Compare repeated searching with several meetings, keeping an unspecified count separate from a stated numerical total.
Emphasizing repeated attempts
何度も, read なんども, emphasizes repetition many times. 同じ場所を何度も探しました。 says that the same place was searched many times.
何度も説明しましたが、まだ伝わりません。 says that the explanation was given many times but still does not get through. Repetition and successful communication are separate facts.
Several occasions without an exact number
何度か, read なんどか, means a few or several times. この人には何度か会ったことがあります。 says that the person has met this individual on several occasions.
何度か試しましたが、まだ動きません。 says that several attempts were made but it still does not work. Attempts do not establish success.
Requesting a numerical count separately
Neither 何度も nor 何度か specifies an exact count. The speaker can emphasize numerous repetitions or report an unspecified set of occasions.
三回試しました。 says that three attempts were made. Its explicit count is more specific than the unspecified count in 何度か.
What count and outcome are supplied by 何度か試しましたが、まだ動きません。?
Show the answer and why
Several attempts were made, and it still does not work. No exact count or successful outcome is supplied.