Here is a link to a very similar problem, I made this a sticky, if you follow this you will note my very first unedited post was the right answer.Also this thread contains a considerable amount of information that you would never find even in the service manual, also note it took Kawasaki techs 6 hours to find the problem in the end. I spent a lifetime trouble shooting electrical everything. Well at 68 I continue to be called in, at times it would be nice if I could download my knowledge onto a flash drive and give it to someone else, someday that may be possible, although we know so little about the human brain.
Very first thing to check is your tip over switch, on the 2013 I think it is located on the left side of the tank. Try disconnecting it giving a few shakes and possibly bringing it inside to dry out. It is one of only a very few things that will give this symptom. Keying off and back on will reset it provided the inside has returned to the normal upright position, hence the light shaking.
The second possibility is a key switch problem, the ecu relay drives the ecu and fuel pump/injectors and stick coils, there have been several cases of key switch contacts failing from moisture, some have been successful in using wd-40, keep in mind these contacts are fused at either 10 amp or 15 amp, so don't put too much faith in wd-40.
Fuel pumps have been known to fail suddenly, the question is, is the fuel pump getting power for the first 3 seconds? This is fairly simple to prove, disconnect the fuel pump, this can be done very easily at the connector on the left side, if you have a meter, test to see if you get power to this plug for 3 seconds on key on. The other way would be to see if the sound with the pump disconnected is the same as connected, which would indicate it isn't being powered.
The last thing to check is frame grounds, especially the ecu frame ground below and behind the fuel tank/ seat mounting bracket.
Some things done by Kawasaki make no sense, one of them being the ECU ground , the ECU depends on the steel frame connection for the negative ground power, in other words, there is no copper connection between the battery negative and wiring harness to the ECU. I and many others on this forum recommend going over all the frame grounds and the engine ground, I use a scotch brite pad and then use a copper coat or electrical grease to prevent oxidation of these connections, as they do get wet and corrode.
So in order of what I think is wrong and place a bet on:
#1 tip over switch @ 95%
#2 fuel pump @ 90%
#3 frame ground @ 75%
#4 key switch @ 55%