From what you've described I may be wrong, but I'm going to guess that you're staying in the Tokyu Stay apartment property..?
If thats true, and if you are locked into staying in their properties, you might want to consider moving to one of their Shibuya locations. I've stayed at their Shin-minamiguchi property for a bit right after it opened, and was satisfied. It takes about 10mins to walk to the main nightlife area of Shibuya from there.
If you have other options and really want to be in Shinjuku or in a couple other of the more lively areas, something like Toyoko Inn might be worth looking into, although for 3 weeks it might get a bit close. It has guest laundry, wired and IIRC wireless internet, decent surroundings, etc.
Tokyu Stay is certainly value for money IMHO, but you have to know what area you are going into. Their Yoga property in, I guess its in Setagaya-ku, and that entire area is quite upscale as I recall.