That book looks interesting, I might give it a go just as an acedemic exercise. The sextant I got for my dad looks like this (except it's all brass)...
Dated sometime around 1910 to 1920 I think, it's had some kind of illumination thing retrofitted to it. I'm sure it'd need calibrating before it was any use.
I've got the book - it has some useful stuff. I bought it because it claims to tell you how to service a sextant - it doesn't. Calibration needs collimators and other optical trickery and is very expensive and completely unnecessary unless you think the sextant has been dropped or massively mishandled. If it has, it probably needs retiring to the mantelpiece as once the arc is damaged the cost of repair will likely exceed the cost of a new sextant.
To check your sextant out is easy, shame you are oop North, as it is much easier to show than describe!
Basically you want to do the following:
1. To check for perpendicular error (of the index mirror) set the sextant to about 35 degrees and hold it out horizontally so you can see the arc and the reflected arc in the index mirror. There should be no step in the arc as it passes from direct view to reflected view. Adjust the index mirror (side screw) until the arc lines up. If it won't line up then the arc is bent or the index mirror frame is not perpendicular to the sextant body. It will look like this:
2. Then check for side error (of the horizon mirror) by setting the sextant to 0 degrees and viewing the horizon with the sextant horizontal or a vertical edge (I often use a lighthouse) with the sextant vertical. The further away the better. You are looking to see if there are two images of the edge, one from the direct view the other from the reflected view. These should be co-incident and I find swinging the sextant from side to side to watch the edge move over the direct/reflected boundary works well. Adjust the horizon mirror until the images line up.
3. Finally of the ones you can do anything about, there's index error. Set the sextant to 0 degrees and sight a far away body, don't forget the shades on both paths if you use the sun. The images should line up. If they don't, you can adjust the index mirror (top screw) to make them do so.
You may find that adjusting the index error throws out the perpendicular and side errors so re-visit these checks. On my sextant it tends not to, but you may find you need to iterate around the loop once or twice especially if the sextant has gone way out of alignment.
Anything less than 3 or 4 minutes of error is traditionally left as on old sextants the adjustment screws work against a solid mounting and can crack the mirrors so you adjust as little as possible - this may be the case on your old sextant so be very careful! On a modern sextant, the opposite side of the mirror from the adjustment screw is held by a spring clip and so you are not creating any stress in the mirror by adjusting it so I tend to adjust for zero error.
The final check is for collimation error and you better hope it's fine because you can't really do anything about it! You sight on two stars at least 90 degrees apart using say the left hand side of the field of view. Then move the sextant so you are using the right hand side of the field of view. If the stars separate, you have a collimation error (the mounting of the telescope is not parallel with the sextant frame).
Hope this helps.