It's very odd how things degrade, especially if your restart the app in between. Klink's doesn't persist any state aside from its app settings, so starting up a connection is basically the same as giving it a clean slate.
Moreover, it's also strange how only some sites are affected, since Klink treats everything the same. Normally, random connectivity issues are due to a flaky USB connection. Your cable is probably fine, as it appears to work fine with your other phone. You might want to see if there's any dust or grime built up in the charging port on your Nexus, as that could lead to a poor connection.
Very odd... I don't really have any other ideas, as I can't recall anyone else having a similar issue. It's usually the case that things work fine or that a poor USB connection/wifi enabled on PC leads to no connection at all!
Kevin