Electrical – Why is the electric wall heater slow to turn on and then quick to shut off

You may have already done this but not sure from your question. When testing the thermostats be aware that you must test the input and output sides. A thermostat is really just a switch. First be sure you have 240VAC on the input sides across the hot leads and 120VAC from each leg to ground. Now connect to the load or element side across the element and look for 240VAC as you turn on and off the thermostat. If you have an AMPROBE, confirm current flow when the thermostat is turned to the highest temp position, indicating current flow through the element, and conversely no current when the temp is satisfied. If the voltages are good on both sides of the thermostat, but no current, it is safe to assume the element is bad. If you do not get the changes in voltage at the load side when operating the thermostat min to max etc. the thermostat is bad. Have you drained the heater? Large amounts of sediment can indicate fouled elements or a partially plugged dip/transfer tube. Be sure to turn off power before draining and refill the tank before turning electricity back on. You can also find a toll free number for tech assistance online for all Whirlpool products and should be able to reach a real person in tech assistance that will help walk you through it. not sure if they will do that for the user, but they do that for us contractors all the time. Good luck.

As I said in the comments to the OP and to Aaron's answer, there are several wiring failures that could cause an NCVT (non-contact voltage tester; usually a pen-looking thing that beeps when you point it at a live electrical wire) to light up, and some that even a plug tester can miss, but that would prevent a "real" load from being plugged in. The first thing to do is get a 3-lamp outlet tester (preferably with GFCI testing) to supplement your NCVT. The tester will correctly diagnose most of these things:

• Open Hot: the black wire is present in the box but disconnected from the plug (or otherwise not making good contact). An NCVT can still light up if the wire is close to the faceplate. However, a plug tester will diagnose the issue. The fix is to flip this outlet's breaker, open up the plate and make sure all wires are firmly connected to the proper terminals on the outlet.
• Open Neutral: the white wire is present but not properly connected to the outlet. Again, a NCVT will light up if the hot is properly connected (maybe even if it isn't), but a plug tester will properly diagnose the issue. The fix is the same as for an open hot.
• Broken/Worn/Bent Outlet Contacts: The wires are hooked up correctly, but the outlet's contacts have poor or no continuity with the plug's blades. You may hear arcing (a rough crackle) as you plug in the tester or an electrical device. A tester MAY show correct wiring at a glance, but if you jiggle the tester the lights will blink off. Replacing the outlet is the solution here.
• Neutral-Ground Swap and High-Resistance Ground: This would tend to affect an entire house's wiring, but it can be localized as well. What is happening is that the white and bare wires have been switched, and in addition, the ground (which normally ties back to the bus strip in the service panel and has full continuity with the neutral) has lost some continuity back to the service panel. There are several situations in which this can happen, most of them involving attaching grounds to house plumbing and then changing that plumbing along the way. An outlet tester may not diagnose this; the expected read is an "open ground" or "open neutral" (remember this would mainly be a problem if the neutral and ground are switched), but if there is some continuity then the tester may not show a problem.
• Poor Continuity Of Hot Or Neutral With Outlet: One or more wires is making very poor contact with the outlet terminal. A plug tester may miss this; there is enough continuity for the tester (which only draws a few milliamps for its LEDs) but even a clock radio would have issues with the load induced by the poor continuity. The solution is to rewire the outlet securely, and if the wiring is sound on the plug then to check other wires in the box (with the breaker off of course). The next most likely culprit is the "bundle" of neutral leads that's normally in an "upstream" J-box; make sure all wires are wound tight and properly secured with a wire nut and electrical tape.