Great Question
I posted an answer to this question when D.U. posted it on ...gasp...Just Helicopters. Topics go rather quickly there, often off the page before they are really hashed out properly. I like the returning to front and center format here.
Anyway, my answer was close to Nick's, which actually makes me feel quite good about myself.
I'll summarize my thoughts here:
The approach was an LDA, but did not specify DME in the heading, therefore DME is not necessary to execute the approach. As stated, GPS can be substituted for DME in terminal procedures provided its an approved GPS with RAIM and etc. So use of the GPS in lieu was approved.
DynamicallyUnstable posed the question that the GPS(DME) failed after identifying the stepdown fix. Once the fix is identified, the altitude is yours. The fix could just as easily have been identified with a VOR radial, which you obviously could have only identified once. In this case the approach used a DME fix, but the effect is the same. After all, the obstacle that you cleared before the fix is behind you. The terps requirements from this point on are still met.
He stated that he started his time. I assumed this to mean that there were alternate ways of identifying the MAP. One would have been with DME and the other with timing. The AIM does not favor one way of identifying the MAP over another. Obviously, DME can be a much more reliable identification of the MAP in space, because it does not rely on steady airspeed control, although legally, they are equal. As long as the approach plate shows a timing box, timing is allowed. You cannot make up your own timing from distance and airspeed calculations, though.
Legally, you can remain at 380' and finish the approach with timing. In practicality, you would have to be confident that your airspeed was reasonably close to desired, however. You would have to notify ATC of the failure and indicate your intentions. It is possible that the MAP can also be identified with radar.
If your airspeed had been all over the place, and you had been relying on the GPS for the MAP, the best action would be to execute the missed approach by initiating a normal climb to the initial missed approach altitude at least, continuing on approach course to the MAP using timing and execute the remainder of the missed approach procedure from there.