These are the original lights in my 1984 house. They are in all 3 bathrooms, 11 total, in soffits above the sinks. They are 7" cans presently with BR30 45W, 65W, or 75W incandescent bulbs. The bulbs are getting to be harder to find, but for now I can still buy them. The cover is a frosted glass dome. The bulb protrudes slightly below the soffit into the dome.
All the LED BR30 replacements that I have found indicate that they should not be used in enclosed spaces. A 65W LED equivalent is only 11W, so I don't think that it will heat up too much. Maybe the life would be reduced somewhat.
All LED can retrofits that I have found are for 4" or 6" cans. It seems that 7" was never very common. Most of the retrofits are recessed, I really don't want a recessed spot, I want a very broad light so it will reflect off the mirror and provide some frontal lighting.
I have found some flat LED round panels that are designed to be mounted directly to the ceiling without a can. These should meet my broad light requirement, and if I can find one thin enough, this might work. None of these are made by the more respectable companies, that bothers me somewhat.
Any ideas? Maybe buy a BR30 LED and try it?