Having worked on many of the MkIX sextants I have written a small booklet that covers all the main problems and shows a couple of calibration methods. This is usually listed on eBay under my eBay ID of Ticktock58.
Without doubt the most common problem with a Mk IX is a leaking bubble chamber. Usually caused by the sealing compound, Shellac, leaking somewhere. In the case of a bubble that cannot be removed, see bubble manipulation, remember, the chamber was probably filled 60 or more years ago so the leak may be very small. In most cases I concentrate on re filling the chamber and do not worry about a tiny leak that will be difficult to find. I invariably remove the cap from the bubble chamber and machine a filling screw into the cap so that re filling is then easily performed. I use standard lighter fuel. The original fluid could be made up but there are problems obtaining the correct fluids. I have even used white spirit. It is thicker and gives a more sluggish bubble but is quite Ok for ground use at normal temperatures.
The next problem to crop up repeatedly is dry, or stiff bearings on one or usually both mirrors. You can try working thin oil into them but frankly the only way to improve the situation is to open the bearings and clean and re grease them. With care both bearings can be lubricated without any special tools. With many to do I made myself some special tools to ease the operation. The main or index bearing is probably the easiest to free off whilst the 5° flick mechanism is complicated in that several operations are performed by operation of the lever. First the mirror is moved. At the same time the shutters over the degrees drums have to be moved to show the different numbers. In the case of the Mk IX that is straight forward but in the case of the versions with the clockwork attachment there is a lever connecting the mirror into the clockwork housing to move its shutter. lossening the bearing can be done but you will need to move slowly and carefully noting the various attachments to the mirror.
Lighting is also a perenial problem. The bulbs used in these sextants are, as far as I can ascertain, no longer available. There are bulbs that will fit, and do look the same, but they do not have the light output of the originals. You need a lamp with 150 mA of current consumption to be any good in the bubble chamber position. The modern lamps are about 50 mA. Bearing in mind that the battery voltage is only 3Volts and with several connections to be made between the battery and the lamp there are likely to be high resistance connections somewhere. Oxidation is always a source of trouble and it may well be invisible to the eye. As a matter of course I always clean, with a small stainless steel or glass fiber brush, any and all connection surfaces. After cleaning I apply Vaseline. This will stop any further oxidation and should stop damage from a leaking battery. The lamp sockets themselves often have intermittant problems with connections to the cables. The wire used is a solid copper and has been formed into a ring under the connecting brass parts. If they look clean I loosen the brass parts and again apply Vaseline with heat from a hot air gun or hair dryer to melt it and enable it to creep into the small voids.