According to Michael Dryden's answer, it takes 877.8W to heat 30ml of water by 70C (from 20C to 90C)in 10s.
My electric shower is about 10 times that power (lets call it 8.778kW for convenience.) I don't like to shower at 90C. I am more than happy if my shower heats the water about half that temperature difference, say 35C (from 5C to 40C). So I would set it to deliver 300ml x 2 = 600ml every 10 seconds. That's 3.6L/minute, so my 5 minute shower uses 18L of water. Of course, if I did choose to raise the temperature by 70C in the shower, I would have to restrict it down to 300ml every 10 seconds, making it a continuous version of your experiment, scaled up by a factor of 10.
The last time I took a shower apart, the element was inside a copper vessel of about 300ml volume. So, if you restricted the flow to 300ml in ten seconds, your residence time in the element vessel would be... ten seconds!
As noted by Schwern, your issue isn't heating it, it's being able to control the temperature accurately. One method that hasn't been mentioned is dunking a calibrated lump of hot metal into it (or pouring it into a hot metal container.) But so far the best method is Michael Dryden's idea of using a microwave.
In industry, batch processes are only used for slow processes like brewing beer. Fast processes (such as those used in oil refining: cracking, reforming, alkylation, etc.) are done continuously. For economic reasons these processes are carried out as fast as possible in the smallest possible equipment, and continuous process is therefore the only practical way of doing them.
If you really need to heat water that rapidly, and the reactants consumed in your experiment are cheap, I suggest you set your experiment up as a continuous process, with a flow of water through it. That way you will be able to control the temperature accurately with an electric heater. A shower will probably be too powerful - try scaling everything down to something like a 25W soldering iron element (make sure it's sealed and 12V DC for safety!)
You can supply your equipment with water from a reservoir at the top and control the flow / residence time with a valve at the outlet at the bottom. If you want to try the same conditions with double the residence time, simply reduce both the water flow and the heater output to 50%.
You may optionally replenish the reservoir with tap water to the point of (near) overflow if you need to mix things into it.
When I did a conversion course from chemistry to chemical engineering, several of the practicals were set up like this. It's definitely the way to go if you want to study fast processes whether they be mixing, reaction kinetics, or whatever.
Regarding Dave's points about critical heat flux, the important thing is to make sure you have sufficient surface area for the volume being heated. I don't see it as a huge issue at your scale though. For 1kW, you need 100cm2 of heating surface for a flux of 100,000W/m2. Many newer kettles provide this (with heating over the whole bottom surface for a neater design) but older models with curly heating elements are above 100,000W/m2, approaching critical heat flux. These electric kettles 'roar' as they come to the boil due to cavitation.
If your reagents are delicate, though, you will need to space your heaters very narrowly, or even use steam injection. I remember one particularly nasty case of electric heaters converting an aqueous polymer solution into volatile aldehydes.