No bluetooth on Gear 2 Neo

I have had a Gear 2 Neo since last Christmas and it happily worked with my then Galaxy S4 and recently with my Galaxy Note 4. Last week I noticed that it wasn't connecting to my phone and investigation showed that the watch bluetooth wasn't turned on (and wouldn't turn on manually). I tried a "restart" which made no difference so then did a full factory reset. This erased everything and took me back to the screens where it wants to connect to a phone but a two finger double tap showed that the bluetooth still wasn't turned on.
I'm pretty sure my watch is toast but I'm wondering if there is anything else I could try.