Thanks for you help Erich. I've managed to get it working now and in case anyone else is faced with this issue I thought I'd document what I did here.
I convinced myself that the device was bricked beyond all repair so organised a swap-out. I felt a bit smug when I plugged it into my W7 machine and the coloured LED was flashing away. So I entered bootloader mode and... it ended up in the same state.
The links you gave me didn't seem to resemble my situation but from them I found this site
DAPLink bootloader update | Mbed and from it I determined my bootloader was simply out of date (and not bricked).
From this site I could download the latest DAPLink bootloader which for my FRDM-K64F was 0244_k20dx_bootloader_update_0x5000.bin.
I was using my old W7 machine so I didn't have to disable the Storage Service. For people using W10 note that the link I give here only talks about stopping the Storage Service but the link Erich gives above also talks about disabling and stopping the Windows Search Service too. I have no idea if you need to stop the Windows Search but I guess it can't hurt.
Step 3 in my link worked fine. I used the link in step 5 (DAPLink ) and entered my board type and it gave me the DAPLink firmware file I needed (in my case 0253_k20dx_frdmk64f_0x5000.bin). I loaded that to the device and when I was finished the coloured LED was flashing again and I now have an updated board.
Thanks for your links Erich. Although they didn't help directly by following links on them I managed to get there in the end.
Regards
Nick