Heater - how to define power needed taking into account air thermal dissipation
Something that will simplify your job is if you can ensure that the thermal connection between your heater and your water is much stronger than that between the heater and the outside environment (through your outer insulating layer). If you can achieve that, then to a decent approximation, all the heat generated by the heater goes into heating everything in your assembly, rather than being dissipated to the outside world, and simple specific heat calculations will be all you need. This would eliminate all the complications of trying to deal with heat transfer coefficients, thermal pathways, and all kinds of thermodynamics.
Of course, eventually the heat leaks out, but if your objective is to calculate the power required to heat your water in five minutes, then this will be a valid approximation if you achieve that relationship between the two thermal paths. Now to find your way to this situation, you still may need to do some work with heat transfer coefficients and thermal conductivity, but you won't need to be all that accurate. You just need to make one number much bigger than another.
As for your heater technology, you will want something that spreads the heat out to as large an area as you can, as you have to worry about getting that heat off the heating element. You could burn it (and other things) up if you can't remove heat from the heating element fast enough. Heat can be spread by conducting it away from the heating element with a high-thermal-conductivity material, or by spreading out the heat generating element to a large area. I don't have specific recommendations there, but maybe someone else could help.
I should add that I am slightly sweeping one issue under the rug, which is the issue of whether the heat can get to your water fast enough without heating your heater up to unreasonable temperatures. This will depend on how spread out you can make your heater, and to really do this part of the design right, you should try to do the heat transfer calculations. But if you can get into this regime I am suggesting, the requirements on your accuracy are a lot lower because you just need to do enough calculation to know that you are in a regime where the heat can get to the water, and you don't need the heat transfer calculations to calculate how much power you need.