I have closed all Chrome windows and used Task Manager to close everything else associated with Chrome but still can't install Flash because it wants me to close Chrome. This is frustrating.
Please ensure you are viewing tasks from all users. Also, it's possible some other process (e.g. chromium-based app) is reporting itself as 'chrome'. Optionally, reboot the system to terminate all running processes.
Note that Chrome does embed Flash Player and there's no need to install the Flash Player PPAPI plugin for Chrome. If you're using Chrome and Flash content doesn't display, see Use or fix Flash audio & video - Computer - Google Chrome Help
If you're using applications built on Chrome (Hangouts), those are keeping Chrome running after you exited the browser.
Also, rebooting is a pretty good solution, if that stuff doesn't all start right back up after restart.