The Boiling or Evaporation of Diethyl Ether
The issue here, as eluded to in your discussion, is that near the boiling point the rate of heat being removed from the ether (via rapid evaporation) was greater than the rate of heat being transferred into the ether, so that the temperature of the ether never reached it's boiling point. In fact, the ether was likely several degrees below it's boiling point due to the evaporative cooling. 

The rate of heat transfer into the solution via conduction from the surrounding air is a function of the difference in temperature between the ether and the air, which in this case was very small. If the ether had been placed in an oven of a sufficient temperature, the ether would have boiled.  

The mass loss that was observed is not at all unexpected, as the temperature of the ether was just below it's boiling point. Imagine how quickly water would evaporate from a beaker if the surrounding air temperature was $\pu{99^oC}$ (and assuming we're at sea level so the water doesn't boil). The vapor pressure of water, and ether, is only slightly less at a few degrees below it's boiling point than it is right at it's boiling point. (Of course the physical action of boiling also allows for more rapid evaporation, so long as sufficient heat is applied to keep up with the evaporative cooling).  

Summary, TL;DR:
As the ether was close to it's boiling point, it was evaporating rapidly, resulting in rapid mass loss. The evaporative cooling near the boiling point happens at a rate to greater than the conductive heat transfer from the surrounding air, so that it never quite reaches it's boiling point, and actually cools to several degrees below it's boiling point.