Jim's mother in law has an iPad and they use it to Facetime to her with auto answering. But they've been having trouble connecting lately. It connects intermittently. One new feature of Find My iPhone/iPad is you can see the battery life. Leo suspects that it may be off the wifi network, and as such, your Facetime account wouldn't see it. But it also has LTE. If you can get a nurse to turn off the iPad and then turn it back on, that will reset all the network settings. You need to do that from time to time. Also, check that it has the latest updates. The MiFi card Jim uses could be powering down and go to sleep. So that might be it as well.
Another option is the Amazon Echo Show. It has a drop-in mode that enables you to video conference immediately. Facebook Portals also work like that, but they are sold out.