Wanhao Duplicator i3 Mk2 Firmware Update
My problem was resolved with the help of comments above.

A quick Google search shows that there are multiple issues with updating to new firmware for printer and LCD (these are separate firmwares!). Useful links are this one and this one which have support for the MkII....
Try to disconnect the display if you go down the Arduino route and see if you get it running while connected to serial (USB) and Pronterface for instance. When pursuing the Arduino route, be sure to send M502 and M500 after the flash to the printer to get the values from the firmware and overwrite the current stuff in the EEPROM.

I downloaded the Mk2 firmware from here and then used the normal Wanhao procedure to update both the motherboard and LCD and this left me with a working printer afterwards. Why the firmware downloaded from Wanhao's own site does not work is a mystery.