I have managed to get working the authentication which i know return a token (JWT) to the client. This token has an expiration date/time so I was thinking about saving the t
Google's authentication coookie is good for 14 days.
http://ben.onfabrik.com/posts/dog-fooding-our-api-authentication
Do not store the user name or password in the cookie. Even if the cookie is encrypted, it is better to store a credential with short expiration time like the token in a cookie than a credential like password which has more shelf life.
Even in the ASP.NET Web Forms or MVC world (Forms Authentication), typically "Remember me" works only until the time the cookie expires. "Remember me" does not mean remember me for ever and there must be a finite time period for remembering. That time can be derived from a cookie. You can put the JWT in the cookie and set the cookie's life time same as JWT, say an hour. When the user comes back to your app within that time, the cookie will not expire and the user is automatically logged in. Otherwise, they have to re-login. Do not think about storing the user name - password and systematically logging in. Let the user enter the credentials and that approach will be more secure. BTW, make sure cookie is encrypted and is an HTTP only cookie.
This mechanism will be similar to Forms Authentication. In place of the authentication ticket, you will use your JWT. Instead of FAM reading the cookie, you will need to have your own HttpModule or a message handler to do that and establish the identity for the requests.