Do you have the space for an adapter like this one?
Lighting EVER Candelabra Screw (E12) to Medium Screw (E26) Enlarger Light Bulb Socket Adapter
Q:
I'm going to overload it by using a larger bulb ?
A:
Electrically you are limited to the maximum amperage that the wiring will support, if it is rated for a 75W 125V then that's 0.6
Amps each (Wattage = Voltage * Amperage). You're not changing the
voltage or number of lamp so it doesn't quite matter (doing the
numbers that is). You can do up to the same amount of actual Wattage
used, which for LEDs would be significantly less.
Heatwise, the LEDs should produce significantly less heat than
incandescent, just make sure you have at least 1/8" between the LED's
bulb and the shade/globe/etc and you should be alright; incandescent
should be more like >1/2".
- CFL Side Note The same electricity rules apply for CFLs (or anything, note CFLs do have a peak turning on but I doubt you'll find a 75 actual Wattage CFL to fit), but I'd increase the minimum gap for heat to about the same as in incandescent ~1/2". I've had a few cheaper lamps with CFLs burn/melt so I'm a little over cautious with CFL and proximity even though they should produce less heat than their incandescent Lumen- relative counterpart.
Q:
The bulbs will be put in sideways; I hear many LED bulbs are unidirectional; what degree/angle should I be looking for?
A:
For unidirectional light, you probably need to find one with a diffuser or multiple LEDs pointing in different directions. If you can see the individual LED(s), it will shine light in the opposite direction(s) it/they are mounted only so if they all point in the same direction it's unidirectional. If you can't see the individual LED(s) then it is probably diffused (though it may not be diffused well)
NO PUTTY!
You have a loose electrical connection and a loose mechanical connection.
Mechanical
First you need to fix the mechanical attachment mechanism. It may be a loose nut on a threaded pipe holding the individual fixture to the base. It could be some other locking mechanism, such as bolts or screws. In some fixtures, the sockets are held in by pressure fits, which are harder to repair.
In either case, you probably need to take at least the lighting unit off of the fan, and maybe the whole fan down, depending on how it is put together. Before you do anything, turn off the circuit to the fan, not just the wall switch or the remote control switch. Once the wiring is exposed, double check that all wires are not hot by using a non-contact tester.
You need to restore the original attachment method, not just a glue or putty to wedge it in place. Fan vibration will almost certainly cause a makeshift repair to break again.
Electrical
It sounds as if you have a loose wire. You need to make sure that all wires, hot neutral, and ground are all connected (in the US, these are black or red, white, and green or bare). The ground may be established by physical contacts among the metal parts, and restoring the mechanical connection may fix an intermittent ground. Sometimes a wire may be under a wire nut, but not well connected. If there is no obvious disconnected wire, remove and check wire connections under wire nuts.
Replacement
If you cannot make the physical socket stable as originally designed, you need to replace the fixture (or at least that part of it). Underwriters approval (a safety standard) is dependent on the structure as designed. If you cannot restore the mechanical structure, you probably need to replace the entire fan.
Best Answer
Hard to tell in a photo, but looks like an E17 to me. Smaller than Edison E26, larger than candelabra. If the base screws are around 17mm, there ya go.
Tricky size to find LEDs in, not impossible but your choices may be limited.