How to quickly reduce the “Troubled by Vermin” quality

Option #1: Oblivion in laudanum.

The solution is extremely Victorian: F.F. Gebrandt's Superior Laudanum, to help you sleep at night. You can find it in the Echo Bazaar (click the "Bazaar" tab), in the shop Nikolas Pawnbrokers. Each bottle costs one Echo, which can be quite pricey if you're trying to sell Whispered Secrets or the like in the Bazaar.

Drinking laudanum is bad for your health, and your Wounds quality will increase a bit one change point), but you can reduce that with F.F. Gebrandt's Tincture of Vigour. I imagine that Gebrandt is an extremely wealthy lady. Alternatively, you can just rest at your Lodgings if your Wounds quality is 3 or higher.

Once you become a Person of Some Importance (PoSI), as with the Tincture of Vigour and Ablution Solution, this option will become less viable; unlike the other two, it will also have a nasty side-affect: every time you drink a bottle of the stuff, you'll become slightly more addicted to it, resulting in the Laudanum becoming less effective.

Option #2: We're all mad here.

Go to A state of some confusion. It will discard and replace all of your Opportunities with new cards, so don't do it if you really like your current Opportunities. Almost all of the Opportunities and actions available in confusion will decrease your Nightmares (though some will abuse your base stats a bit, and one or two will increase your Nightmares while giving you something else you want), and you will leave A state of some confusion once your Nightmares drops to zero.

There are two ways to get to A state of some confusion:

1. Go to Mrs. Plenty's Carnival, into the House of Mirrors, and then Gaze Into Dream's Mirror, which requires that you have at least one in Nightmares.
2. Let your Nightmares quality increase to 8. You'll be taken there immediately.

A state of some confusion doesn't cost you any Echoes, but has a much worse action-efficiency compared to drinking laudanum. (Laudanum seems to do about 4 change points of Nightmares reduction, and actions in a state of some confusion are more like 1-2 change points of Nightmares reduction.) Not all actions there will decrease your Nightmares, but drinking laudanum will waste some of your time with curing your Wounds, so it's a wash; there's some good reasons to just let yourself go mad. Unfortunately, a few of the actions in confusion have a chance to abuse your four main qualities, and there are a few other small-ish drawbacks, so it's not all upside.

Option #2 if you have Memories of Light: we're all mad here, too.

If you have 10 Memories of Light, then you can spent two of them to go to the Mirror-Marches instead of A state of some confusion at the carnival. If you have at least one Memory of Light, then getting to 8 in Nightmares will take you here instead. The Mirror-Marches are much more generous than A state of some confusion; your main qualities won't be abused nearly as much, and actions that remove two or more points of Nightmares are much more common.

Option #3: Psychotherapy.

In your Lodgings, you'll get an option to Find a way to deal with your Nightmares if your Nightmares quality is 4 or higher. Spend a Sudden Insight (possibly gained via Invite a friend to join you in something terribly intellectual from your Lodgings), and you'll lose 5 change points of Nightmares. Your friend will gain one change point of Nightmares and also spend one action, so you should return the favor sooner or later.

The most efficient way: the right storylet

Find a storylet available to you that rewards you in Foxfire Candle Stubs. Storylets reward you in direct proportion to the minimum main quality required to unlock them, so find the hardest one that you can succeed in on a fairly-regular basis. (For example: a storylet that requires Watchful 20 might reward you with 20 Foxfire Candle Stubs, but a storylet that requires Watchful 50 might reward you with 50 Foxfire Candle Stubs.)

Alas, I can't be more precise than that: most of Fallen London will be unavailable to you at any given moment, because one of your four main qualities will be too high or too low. Storylets are only available for a particular window: if your Watchful is too low, you don't get to see, say, Search For A Missing Rubbery Man in Ladybones Road. Similarly, it will disappear again once your Watchful gets too high.

In general, storylets involving Knife-And-Candle players, New Newgate Prison, and the Church will get you Foxfire Candle Stubs.

The fastest way: Money, Dear Boy

If you really need a lot of candles fast, then the simplest answer is to sell things, and buy Foxfire Candle Stubs at 3 pence apiece from Merrigans Exchange in the Echo Bazaar. (It's the "Bazaar" tab.)

Anything in the "Goods" category is ideal to sell, because those items can't be converted into other things and doesn't have any use except to be spent. Rostygold, Moon-Pearls, Nevercold Brass Slivers, and other such Goods have uses in storylets, but can't be up-converted into other things. Chances are really high that you have far too many of several different types of Goods, so sell off the excess: remember how many you used the last time a storylet required that Good, and sell off all of that Good except for a small reserve. Repeat for the rest of your Goods, until you have enough Echoes to buy what you need from the Bazaar. If you can't or don't want to sell Goods, then sell the base-level up-convertible things, like Whispered Secrets, Jade Fragments, and whatever else that you've ended up with far too much of.

The main idea here is that you will almost never have 1000 of the thing you need, but you'll almost always have 2000 various other things that you aren't using terribly much at the moment. Since you can't really predict what you'll need next, it's a better use of your time to buy what you need now, rather than to save up forever in the hopes that you'll eventually find a storylet that requires those 10,000 Whispered Secrets that are cluttering up your cupboards.