For a starting point, consider the meat aisle at the grocery store. In it you'll find roasts, steaks, and other chunks of red meat. This government food-safety site says that such meat can be kept in the refrigerator (40 deg F) for 3-5 days. That's for meat that's already been cut up, drained of most of its blood,1 and packaged for individual sale. Before that, it could have been kept for a couple weeks or more. That's just some context; stuff that came from dead livestock, in general, can be safe for rather a while.
That's for meat that people are trying to keep safe. But that's not Dracula's case; he's not raising livestock. So while bodies kept at the right temperature could supply some blood for a couple weeks, it's bounded by how long a body not kept at the right temperature can sit around before the blood is dangerous to consume. We also have to consider coagulation -- how long until the blood isn't sippable any more because it's clotted?
Spoilage due to bacterial infection in the blood can begin within hours, but appears to come from two main sources: bacteria in the intestine at the time of slaughter, and infection caused by the slaughtering process. Indeed, one government agency has special guidelines for collecting blood for human consumption (188.8.131.52.1 ). Dracula should select for bodies that died intact, not roadkill or ones that died from wounds, to minimize risk of infection.
As for coagulation, if there has been an opening (like an incision), blood coagulates quickly after death (within minutes). If there is no wound, however, blood remains liquid for several days after death. (See also the Wikipedia article on lividity.) That's true even at room temperature. (Freezing does bad things in the coagulation department, but that isn't Dracula's situation.)
Putting all this together, an intact body with no special precautions will contain liquid blood for several days. If it is kept cool during that time it should be safe on the infection front. (Plus, maybe vampires don't worry about infections; consult the physiology specifications for your particular model of vampire.) If he cares about blood-borne infection, Dracula will do best in cooler climates, whether natural or man-made (morgues).
1 I'm going to draw on an unusual source for this: Jewish dietary law. There is a religious prohibition against consuming blood. When an animal is slaughtered its blood is drained in the usual way (just as it is for other slaughter), but Jewish law requires an additional process of salting the meat to draw out more of the blood. It's of course possible that this is completely redundant and what the rabbis who instituted this were seeing wasn't blood but water with myoglobin (thank you Selenog for teaching me about myoglobin in meat). But I've heard a lot of arguments over the years about how such-and-such practice shouldn't be required any more because science, and I have never heard an argument against this one.