The switch is crap, not to put too fine a point on it.  It consists of a cork float inside a metal sleeve, connected to a silver-plated disk that's supposed to make contact with a couple of silver-plated contacts.  The cork soaks up fluid and sinks, or expands and gets jammed within the metal sleeve, or the contacts get corroded, on and on and on.  The switch either won't work when you need it to, or it keeps the warning light lit all the time so you either learn to ignore it or remove the bulb or cut the wires.

The problem with all this is that this switch is an important safety feature.  If, for some reason, all your brake fluid leaks out the bottom of the car, it'd be far better to learn about it from a warning light than by pressing on the brake pedal and getting no results.

Sure, it's possible to fix the switch -- but it's not possible to fix it for long.  It's crap, I tell ya.  The solution here is to replace the switch -- and the most workable way to do that is to replace the entire remote reservoir with one with a good switch in it.

Fortunately, there are such reservoirs available from other types of cars.  Japanese cars tend to be preferred here, because they come with magnetic reed switches that are indestructible -- they will work, period.  Walter Acker IV found a very suitable remote reservoir in several models of Mitsubishi.  You can visit your friendly neighborhood junkyard, or you can save yourself the junkyard diving and just get a brand new reservoir -- complete with the diaphragm fix mentioned below already incorporated -- from any Mitsubishi dealer.  The part number is MB534534, and costs less than $40.  This part number includes the reservoir itself, cap, diaphragm, float, and switch, but not the mounting bracket nor the hoses.

The Mitsubishi reservoir is roughly the same shape and size as the Jaguar reservoir.  However, it doesn't mount the same way, so a bracket must be fabricated.  It also has 10mm hose connections, considerably larger than the Jaguar's 1/4" connections, so adapters must be used to connect the hoses.

Here's the original reservoir in Kirby Palm's '83:

That rubber boot on top conceals the connections to the crap switch.  The dimple in the center is where you can push to force the plunger down to test the warning light circuit.

Note that the hoses connecting this reservoir have already been altered.  The original arrangement involved a pair of metal tubes with four short pieces of hose and eight clamps, but Palm removed all that and simply connected long hoses directly from reservoir to master cylinder.

The support is an arm attached to the power brake booster mounting bolts.  Here's what the support looks like with the reservoir removed:

Y'know, spilled brake fluid is really hard on paint.

And here's the bracket used to mount the Mitsubishi reservoir to the original support:

The Book contains a plan for this bracket; actually, the plan in the Book has been upgraded to include a tab for holding the electrical connector, but otherwise it looks just like this bracket.  You'll also need a 3" hose clamp.

Here's what the Mitsubishi reservoir looks like installed:

The switch on the Mitsubishi reservoir is on the bottom, completely outside the reservoir.  Only the float is inside, with a magnet in it.  When the magnet gets too close to the bottom, the switch turns on the warning light.  Every time.

Bry Schreurs did the same job a different way.

Now that you've done such a fine job installing the Mitsubishi reservoir, you find out that it has a problem too!  Fortunately, its problem is not life-threatening, and since we're talking the Japanese here rather than the Brits, the fix is available at any Mitsubishi dealer.

Walter Acker IV found that there is another brake fluid reservoir that can be used in the XJ-S.  It's also from a Mitsubishi, but from a pickup truck.  It's a bit smaller and rectangular, and has a black cap instead of white.  It this case, it might be helpful to get the bracket with it.