http://www.victorianlondon.org/ is packed with wonderfully vivid primary sources. If you used a book like “Victorian London” by Liza Picard to get an overview, then read through the bits of this site that appealed, I think you’d have an excellent sense of the time and place.
Here’s what the site has on rain, for example:
He who has not seen it rain in London, has not seen London; and I had this pleasure the morning I went to see the Tunnel under the Thames. Then I understood how, in such weather, one can be seized with the temptation to give one’s self a pistol shot. The houses drip as if sweating; the water seems not only to descend from the heavens, but also to ooze from the walls and ground; the sombre colors of the buildings turn yet gloomier and take on an oleaginous look; the beginnings of the streets seem like entrances to grottos; everything seems foul, used up, mouldy, and sinister; the eye knows not whither to turn, not to meet something disagreeable; one feels shudderings, which have the effect of a sudden attack of misfortune; one feels an irksome sense of weariness, a disgust with everything, an inexpressible wish to go out like a lamp from this weary world.
Edmondo de Amicis Jottings about London (trans), 1883
It also has plenty of maps like this one, from 1899: