Investigate reverse engineering and reimplementing amazon's OAuth2 Flow
Looking at the general issues that people are having with the plugin it seems like the vast majority revolve around people having to login multiple times per week/month and that is majorly hindering the experience.
Currently the login code perform a login and stores the session cookie. This cookie appears to expire either due to a fixed expiration date or potentially due to a lack of keep-alive or activity from an unimplemented API.
A better solution would be to investigate how the Alexa iOS/Android app is performing its OAuth2 flow to get a refresh and access token, then simply use the access token if not expired or refresh the access token with the refresh token if the access token does indeed become expired.
This will likely require some major refactoring for logging in, but I believe it would overall provide a better experience and ultimately the refresh/access token flow should allow the alexa_media_player to stay logged in and functional as long as the refresh token remains active, which should be a very long time.