You should have a storylet in your lodgings titled "Investigating the Big Rat", to progress this you will need rat companions.

Rat companions (like the Talkative Rat, Bandit Chief, etc.), the more expensive, the better they are at generating Having Rodentine Minions Investigate, and their luck checks will be easier.

To progress further, you need to get HRMI to 5 to unlock accompanying storylets (just Perhaps Not out, they should be right next to the main storylet), each option there will need a rat companion and a certain amount of HRMI depending on the rat, the more expensive, the more needed; the check will also be a Serenity of the Plaster Face check.

To get the Serenity of the Plaster Face up, you will need to sacrifice foxfire candles and lamplighter beeswax in the main storylet to increase it.

Keep doing this for levels 3-8, at level 9, you will need to lower the Serenity to less than 5 to progress, the storylet for that stage will allow you to sacrifice phosphorescent scarabs to lower it by -6 CP.

After this is done, you will need to build up running battle... to about 10, then travel to the Flit, there, a storylet should unlock that lets you decide the ending.

Handing it over to the constables or killing it will give you the Albino Rat.

Playing with Broken Toys can be progressed to 10 after playing its card with the Albino Rat and Seeing Through the Eyes of Icarus 15.