Sweeney Todd can be run easily as a horror story, or as a monster on the rampage, depending on both how it's approached and which side the PC's are on.
The two most closely suited for the purpose are Ghosts of Albion and Cthulhu by Gaslight. These two have advice on running games of investigation into horror, and Sweeney Todd is a suspense/horror borderline; both games are set in the victorian era, meaning little to no conversion needed. Both games have supernatural sources in mind, however, but in both cases it can be ignored easily. (Ghosts of Albion is similar in setup and tone to BTVS, but less genderist and moved back 100 years... and co-created by Amber Bensen, one of the Buffy cast...)
The use of Castle Falkenstein is also a viable option. While it lacks specific advice on running investigative games, it's designed to be very well focused upon victorian setting issues. From the skill list to the card based engine, it aims for immersion. The default CF setting is a parallel earth, with some non-humans (Faeries, dragons), superscience and magic, but those can very easily be ignored.
If one is already familiar with GURPS, Call of Cthulhu or Savage Worlds, the setting books for those systems are viable choices - players have far less need to be familiar with the rules than the GM, so GM familiarity can make the game smoother - but generally, the genre emulation is often not mechanically assisted in such conversions.
Spirit of the Century is early 20th century, but the mechanics are excellent, simple, and adaptable.
Ravenloft is adaptable if the only system you know is D&D, but it really doesn't do the era emulation at all mechanically, and is mostly survival horror. The White Wolf version is better able to do it, because d20 is more flexible than AD&D, but it's still going to be mechanically a poor fit.
Vampire by Gaslight is a set of LARP rules modifications for playing Vampires in Victorian settings. The system is quite useable, but focuses on vampires, not humans.
- The best known Victorian RPG systems:
- Space 1889
steampunk science fiction elements added
- Castle Falkenstein
has fantasy elements added
- R Talsorian Games
playing card based engine, standalone rulebook
pdf and OOP print, stock number CF6001
- SJG GURPS version GURPS Castle Falkenstein
Setting Book for GURPS system
print only, OOP, stock number #SJG6099
- Cthulhu by Gaslight
setting book for Call of Cthulhu.
- Some lesser known Victorian RPG systems:
- Vampire by Gaslight
White Wolf - Old World of Darkness Minds Eye Theater setting book.
Live Action Role Play - Mind's Eye Theater system
Note: this focuses on playing victorian vampires, and is probably unsuitable.
- Ghosts of Albion
Eden Studios - Standalone Core Book using Cinematic Unisystem rules (d10 only)
based upon the TV series of the same name.
- Not quite victorian
- Spirit of the Century
Evil Hat Productions, Standalone corebook using FATE engine (dFudge or d6 needed)
setting is 1920's style adventure stories. Easily back adapted to Victorian.
PDF, Print & PDF bundle
- Ravenloft setting for AD&D or 3E
TSR (AD&D) or White Wolf (D&D 3E); WotC 4e version scheduled.
setting is not truly victorian, and very much horror-fantasy.
There are some other indie games focused on investigation, and some other victorian games, but not being familiar with them, I can't recommend for/against them.