I don't think there is any point using an Arduino for this at all. The ESP8266 has a fairly powerful microprocessor in it and it's perfectly capable of doing this without an Arduino. Not only will that simplify the whole design, it will make it much smaller.
You are right not to include a power button because this is something you don't want to be turned off by accident. I also think you should only have one large button to avoid the risk that she might press the wrong button. If she were to fall and hit her head for example, she might be confused and press the wrong button.
The ESP8266 draws quite a high current when it's connected to wifi so you want this device to be in deep sleep mode most of the time. It should wake up when the button is pressed, connect to wifi and then communicate somehow with someone who can help. Twitter would probably be a simple way to do that so you want to use a web service like IFTT to send a tweet when the button wakes the device up.
If the device is in deep sleep mode most of the time, it could run on two AA batteries for a very long time. You want it to wake up occasionally just to check that it's working. Depending on how frail your grandmother is, you may only want it to alert to let you know something is wrong and just to communicate that it's still working.
If she is very frail, you may want her to communicate more regularly. You could still do this by making it beep every few hours, lighting a green LED and explaining to her to press the button when this happens to confirm she is OK.
Decide how you want it to work and how it will best fit your grandmother's needs, then I'm sure people here will be very happy to help to put it together.
This sounds like a really good project. Whilst the link below doesn’t exactly do what you want it might be worth a look as the PIR could be replaced with a momentary push button (or buttons). This however should not be relied on in a life threatening situation. As a project for reassurance I would be happy to help if I can.
viewtopic.php?f=11&t=4458&hilit=pir