Find/add the Airport Express in the Home app.

A 2nd gen 802.11n with v7.8.1 firmware would be considered an AirPlay 2-enabled speaker. As such, you should be able to add it to the Home app as an accessory, as follows:

1. On your iPhone, tap on the Home app > tap on Add Accessory > tap on Don't have a Code or Can't Scan
2. When the AirPort Express appears, tap on it to select it.
3. Wait for the Express to be added.
4. Provide the Express a name and choose which room you want it associated with.

If the Express does not appear, then check the following:

• Use a base station name that is 16-alphanumeric characters or less that does not include any special characters or spaces.
• If the Express is NOT your main wireless router, configure it with a static local IP address.
• Configure the Express to use "IPv6 Link-local only." AirPort Utility > Select the Express > Edit > Advanced > IPv6 > Configure IPv6 = Link-local only

Ref:

• AirPlay system requirements - Apple Support
• Add and manage TVs and speakers in the Home app - Apple Support