Well, you can't.
If few weeks is all you have, and there is no UX related research, there is little you can do without missing anything important.
Breadth rather than depth
You'll have to try to focus on breadth (the surface level) rather than depth, since the deeper you go the more research time you need. For instance, to decide whether an action should take a coloured button or a link requires knowledge of how frequently it is used, or how important it is for the users.
From personal experience, all the personas business, user-testing, task modelling go out of the window when you are required to do general work on large systems within a short time-span.
Localised improvements can reduce satisfaction
What's more, improving substantially a small part of the system is going to put the system in a state of imbalance, which can actually upset users. If part of the system has exceptionally high-usability compared to other parts, users will be annoyed by the quality of the other parts. Whereas if all the system involves the same level of low-usability, users are more in peace with it.
This may sound highly dubious, but intensity ranking or impressions (such as how easy to use the system is) are always relative. If within the system you provide a reference with high-usability, it immediately makes the low-usability parts of the system more noticeable. In the way our brains work - we recognise differences (or contrast) much better than absolute values (in fact, as far as mental cognition is concerned - there is no such thing as absolute values).
Exhaustive UI review?
Anyhow, while it's hard to tell without more details, I guess one of the logical courses of actions would be simply to conduct and exhaustive UI review; meaning, going page by page and see how the arrangement, visual design, and interaction patterns can be improved; and doing all this based on knowledge of simple UX heuristics and cognition. For some decisions you may to consult the stakeholders (how important is this, etc.), if not only for the sake of impressing and showing them that the UX process is not one done without knowledge of experts.