I'm mostly with dragonchild on this one. Most systems, especially software, tend towards anarchy over time, and you are the one who has to deal with it. The more you give into "just kludging it", the harder your job becomes in the future. In fact, it sounds like that same attitude on the part of your predecessors is what created this situation in the first place. I wholeheartedly believe your work should bring the system to a greater place of stability (within a reasonable amount of work effort, since you're still trying to maximize profit). It's almost never the case that long term profits are maximized by letting your systems slide into chaos.
in reply to On contexts, rules, flexibility and purity
That said, I understand there are business situations where kludges might be more desirable then not delivering a product at all. I'm not saying you should never do them. But generally management doesn't understand the difference between a kludge and a good long term solution, except that the latter usually costs more money. If you have to write a kludge, write one that at least pushes the system in the direction of order rather than more chaos.
In other words, what kinds of solutions can you write that either A) bring you closer to the ideal solution or B) are absolutely temporary solutions that will break soon enough that management realizes a more ideal solution is also more cost effective?