Extruder Heater Failure

Are you sure it is your heater and not your thermistor?  Seems like there have been lots of thermistor problems here recently.

When I was having intermittently failing thermistor readings, it acted exactly like this. It would drop to readings of 170 then go back to where it was then go to 140 and back and so on getting progressively worse. I still use pronterface since it runs better on my old acer netbook though. I sometimes got mintemp warnings and sometimes didn't. Replacing the wires only solved it for me. However, none of this means its not the heater. Just an easy thing to check and eliminate as the cause.

There was a wiring problem between the thermistor connector and the thermistor?

I reset the printer and tried to heat up the extruder again.  Now I see lots of power used, but no change in temperature (either on the graph or by touching it with my finger bones).  I've never understood all of the lines and colors on the RH graphs.  If the green is power, then it is trying to send lots of power to the extruder, but none is getting there. I guess I'm about to disassemble the hot end, eh?  Any words of advice?

Yes. You have to hold the little clip out of the way, with your thumbnail.
Even so, it takes a little wiggling and pulling.

(If I remember right, the clip pulls down just a bit, and releases the white connector.)

There should be a clip on the connector itself that holds the connector together.  You have to get the clip out of the way to pull the connector apart

The connectors I would imagine would be fine. Or at most the connector on the extruder side, but I doubt that. The board side see's almost no movement, so again most likely fine. If the wire has failed it is most likely at the point with the largest kink in the loom. So something like all the way left at the back, on my SD it kinks the loom very harshly near the Extruder Motor heatsink & fan at that position. If you are going to get a dry joint, it is most likely going to be there.

I would cut both connectors with about 1 inch of spare cable, strip yourself of exposed 4mm of wire. Check for continuity between the connectors and freshly stripped end. If they are fine, get some heat shrink, new wire, solder and replace the mid section. Test to see if it works, then either cable tie it to the existing loom, or fight your way through the conduit like stuff. (I'm a little lazy and just cable tied my G-Code Fan wires to the loom)

If the connectors gone, I would imagine something similar to these would do the trick RC 2.54mm connectors

Ian or the others may have different advice.

Depends if the insulation is fine, and it is a dry joint within the wire. Bit more piggish to find.

I imagine that engineering cables that flex is a solved engineering problem.  I also imagine that the SD approach is adequate for the short term, but perhaps not adequate for long term and heavy use.

I wonder if using a ribbon cable would improve things.  It might not be easy to bend in all directions, however.

Coming in from the top center would be a good idea, but would you need to make some conical/pyramidical structure to serve as the cable's origin?  How would that interact with the desire to enclose the printer?  I am imagining a coil of wires, spring shaped, so that each part of the wire flexed minimally.

Ink jet printers use flexible cables, but they only flex in one direction (back and forth along one axis).

You could buy a pack of Pololu jumper wires and just replace the wires as needed.  It would not be very difficult or expensive.  Or maybe someone will start making and selling complete wire bundles to swap in quickly and easily.

This should be an interesting discussion.