So who takes over if Kitzhaber resigns?

[Bryce Butler]

6. Kate Brown would appoint someone to fill in for the remainder of her term...

From the Oregon constitution:

Governor to Fill Vacancies by Appointment

When during a recess of the legislative assembly a vacancy occurs in any office, the appointment to which is vested in the legislative assembly, or when at any time a vacancy occurs in any other state office, or in the office of judge of any court, the governor shall fill such vacancy by appointment, which shall expire when a successor has been elected and qualified. When any vacancy occurs in any elective office of the state or of any district or county thereof, the vacancy shall be filled at the next general election, provided such vacancy occurs more than sixty-one (61) days prior to such general election.

- Article 5; Section 16