Hmm, how detailed information do you need? If you just want the memory usage you can ask the GC. It knows. ;)
long bytes = GC.GetTotalMemory(false); // use 'false' to not wait for next collect
The variable 'bytes' will contain the number of bytes currently allocated in managed memory. I'm not sure whether the managed memory entails the entire process or just the AppDomain. You'll have to test this by running several AppDomains in one process and see if managed memory allocation is measured cross AppDomains. If they don't, then you can use this to measure total memory usage in an ASP.NET application.
If you want more specific information there's a diagnostics API for the CLR which you could interface with. There's also plenty of memory profilers out there, but if they'll work within an ASP.NET application I cannot say.