Halogen to LED conversion on multi-light fixture not working
The problem is likely a mix of old and new technology: the power supply for the old halogen incandescent lamps is a newer electronic converter, rather than a transformer. Some of these converters have a minimum current for operation, which the LED's don't meet.

You have a few options:


Replace the existing power supply with one designed for LED lighting (This is a generic example; choose one that meets your needs).
Run with two of each type. You'll save some energy over all-incandescent, but not as much. In addition, the difference in color temperature may be an advantage -- a more complete spectrum, or a disadvantage -- noticeable mismatch in color.
Try a different brand of MR16 LED lamp... but there's no guarantee it would work better unless the lamp is specified to work with the particular model supply in your fixture.