Which is the best AES encryption dll for C#

First advice is to never store anything sensitive in your code directly. You can always reverse engineer that, no matter how cleverly you try to obfuscate it.

I've read about things like breaking a password into several pieces, placing them at different places in the code and running them through a series of functions before finally using them... although this makes things harder, you can still always monitor the application using a debugger and ultimately you will be able to retrieve the secret information.

If I interpret your scenario correctly, what you have is code that is to be deployed at some client's premises and your code is connected to a database (which I suppose is also under the client's supervision), connecting to it requires a password. This password is known to that client, so trying to hide it from the client is rather useless. What you do want is to restrict access to that password from anybody who is not supposed to know it.

You typically achieve this by putting the sensitive information in a separate file in a folder that should have very restrictive permissions, only the application and a handful of selected people should have access. The application would then access the information when needed during runtime.

Additionally encrypting the separate file turns out to be a problem - if you do so then there is a key involved that again would have to be secured somehow - infinite recursion is on it's way :) Securing access to the file is often sufficient, but if you really require to be as secure as possible, then a solution is to use password-based encryption for the file. But the idea here is not to store the password in yet another location on the system, but rather as out-of-band information (e.g. in a physical vault) and entering the password when starting the application. This, too, has its problems: physical presence of a person is required for (re-)starting the application, and you could still retrieve the password from the RAM of the machine where the application is running on. But it is probably the best you can do without specialized hardware.

Another good alternative to password-based encryption would be to rely on OS-specific "password vaults" such as Windows' isolated storage it's sort of a trade-off between not encrypting at all and keeping the password out-of-band.