Open your browser and navigate to the link below and check the IP returned before and after enabling your VPN software. You should see your VPN provider's IP when VPN is enabled to route your stuff via VPN provider's server. Otherwise, you'd see your hotel wifi provider's IP.
Update:
As you know, the above check verifies if your browser is going through the VPN tunnel or not. So if all you care about is your browser traffic, this check is good enough. However, if you want to absolutely make sure all your traffic is routed through your VPN server, you can check your kernel routing table to make sure it is setup correctly by your VPN software which I am pretty sure is done correctly.
The following list is part of my kernel routing table. As you can see, my VPN inserted routes to override the 'default' route which is on my physical interface (i.e. en1) to VPN tunnel interface (i.e. utun2). As long as you see similar things on your machine you can be sure all your stuff is going through your VPN tunnel. I do not know what OS you are on but the following check can be done on MacOS or Linux and I assume Windows has similar tools. Open a command shell and do the following.
arul@lion$ ip route list
0.0.0.0/1 via 10.8.0.1 dev utun2
default via 192.168.1.1 dev en1
128.0.0.0/1 via 10.8.0.1 dev utun2
...
You can also do the following additional check that tells you how your connection will be routed. In this example, it tells me it would use my utun2 device to get to a public IP i.e. 8.8.8.8
arul@lion$ ip route get 8.8.8.8
8.8.8.8 via 10.8.0.1 dev utun2 src 10.8.0.75
Finally, you want to absolutely make sure you are only using VPN tunnel, you can delete the default route which I am pretty sure is how your VPN software implemented their 'Kill switch'.