I regularly use the Pasha Resort Hotel, which have nice rooms and easy location. I was pleasantly surprised recently that in addition to the free mineral water, they also had a can of ラムネ. Probably special summer treat.
Recommended with two warnings: (i) do not go for any rooms on the 1st floor (near the reception). mobile phones don't work there, no coverage. (ii) 2-3 of the smaller rooms do not have a fridge with cold drinks, only pot and cups for hot drinks.
Others I would recommend in Shinjuku are Hotel Pasha and Hotel Forsion from the same group, though they are slightly a bit more expensive, and some DHs charge 1000-2000 yen extra for that location (considered Shinjuku, not Kabukicho).
Others I've heard good things of (from providers) are the Bali An LHs, though they are always full when I go (long wait even for チェックアウト待ち) and the lobby is very open, like a normal hotel.