Notification when a special AppleID/device connects to internet?
By default, iOS devices only connect to Apple's push notification servers when they connect to the network, so you would need to interact with that system as a paid developer or consumer of a MDM solution that allows you to send notifications and retrieve status when messages are sent and received to that phone.

It's not a hard engineering problem, but you need to get software installed on her device by jailbreaking or by means that Apple approves. An app that registers for a local notification and attempts to send a http request to your web server is all you'd need to track each poll interval that she has an internet connection.

Realistically, you will probably end up asking here to reply to a message you send and trusting that iMessage will send a "delivered" message to your phones promptly after her phone registers to the APNS servers and gets the messages that are queued for her device/account.