How long does the Feast of the Exceptional Rose last

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 reliable way to gain access to The Shuttered Palace is through the "Open a way to other parts of the City" storylet that you can find by visiting your lodgings:

To unlock this option, you need 400 bottles of Greyfields 1882, and 1000 bottles of Greyfields 1879.

A less reliable method is to wait for a "The Ways of the Shuttered Palace" Opportunity card to appear, which will provide you with a few options:

This card appears for those with a Persuasive skill of 40 to 80, and offers a "day-trip" to The Shuttered Palace for 50 Silk Scrap and permanent access if you are willing to spend 5 Fate (or Nex, as the case may be).