Wake on LAN is a LAN feature, not WAN, so you’d need to issue that over the local LAN there at the house. You’re going to have a hard time trying to get that working over the WAN (if that’s even possible).

The other comments mentioning a scheduled boot would be a much easier/simple solution if it works for you.

But I’ll throw this in, the super basic least tech solution to this is to open a port forward to the house’s network router. Yes, I know you don’t want to do that, but it’s probably the only network device at that house that’s actually on 24/7 right? And by all means lock it down however you like. My simple method is to open the router login to a non-standard port number, with a IP whitelist, add my own home IP address to that IP whitelist, and bam you now have access to that remote home’s router for just your IP address. Log in remotely, issue a wake on LAN via the router’s own web ui, done.

It’s perfectly reasonable to make this a bit more secure if you wanted but it gets slightly more complicated - open a non-standard port for SSH access to the remote router’s SSH port that only allows SSH login with key. Generate a SSH key and share that key with yourself, then you can log in remotely to that remote house via non-standard SSH port using the SSH key (no user/passwords). From there you’d have to see if you can issue Wake on LAN on the SSH command line, or set up a SSH tunnel from that remote LAN to yours so you can proxy into the router login page and do your Wake on LAN from there. … yes I realize this got complicated :/ But you’ve got a few things to explore given your patience for tinkering with this stuff :)

Of course much of this relies on that house’s router having any of these features to enable and configure. The main takeaway here is that Wake on LAN requires something on 24/7 at that remote LAN for you to enable remote access into and issue a Wake on LAN command within that LAN. How to actually accomplish that is the tricky bit.