Unless you know a lot about Windows internals, trying to
specifically debug something like this is probably out of the question. Even then, it would have to be an exercise in reverse engineering without access to the LoL client source code and a debug build.
For your purposes, I would recommend Process Explorer from the
Sysinternals Suite (now absorbed by Microsoft). It's probably the most thorough general-purpose task manager available on Windows.
Another useful tool might be perfmon.exe, which I believe ships with Win 7. It's pretty much the way to go if you want to profile process or system metrics, since it uses the system profiling API to get high-precision timer data, etc. Just be aware that in order to do its thing, perfmon has to enable a bunch of process and thread accounting ... stuff... in the kernel which doesn't normally run. In other words, having perfmon running will itself degrade system performance. Generally speaking, the more counters you have running, the worse it will get.
There are even deeper tools in the Sysinternals Suite toolset (like procmon) but as I said, you at least need general familiarity with the Windows API to even roughly grok its output. Supposing that you first built up a reasonable case for a memory leak using high level tools like procexp and perfmon, you might be able to use procmon to hone in on exactly which thread was leaking memory, so that you could report back something more specific and helpful to the developers than just "I think you're leaking memory".
Your problem does sound like a memory leak except for one thing that strikes me as odd. It shouldn't be necessary to reboot if all that's going on is a user space memory leak. Once you kill the process, any and all pages allocated to it are reaped by the Windows heap manager. That is, you get the leaked memory back. The only way that a process could leak memory beyond its lifetime is if it's leaking kernel memory. Does the LoL client install a kernel component? I can't imagine why it would, but other than a bug in the Windows kernel itself or a kernel module (i.e. "driver" in Windows parlance), that's the only plausible way it could leak kernel memory. I mean, I'm as cynical about Windows as the next guy, but this isn't Windows 95. Also, if it is a kernel memory leak that persists beyond program life, it wouldn't just be the LoL client that slows down. Once physical memory is exhausted, the entire system should come to a crawl until the next reboot.