I am a big fan of hotel wifi (when it is fast)!
Most modern hotel rooms have Ethernet jacks. In our experience the Ethernet jacks in hotel rooms are very rarely secured in any fashion and you can simply plug in a device and go.
Although its rare, if the hotel does have a login/authentication splash page for Ethernet users you can simply visit it with one of your devices (while connected to the wi-fi router). The router has a single IP/MAC address and doles out access (just like your home router) to all the attached devices. Once you accept the terms of agreement with one device through the unified access point the security system will let anything connected to the portable Wi-Fi router, including the Chromecast, pass through just fine.
Read this interesting article from WiseGeek, How Can I Use My Google Chromecast In a Hotel Room.