Home ethernet - daisy chained

That's not wired for Ethernet at all. Don't be fooled by the presence of an RJ-45 and Cat5 cabling. They wired them both for analog telephone even though only one of them is RJ-11 (you can plug an RJ-11 telephone line cord into an RJ-45 jack and it will fit "okay", and make good contact with the middle pair or middle two pairs). They're using the green pair for line 1 and the blue pair for line 2 (or vice-versa).

You're almost certainly going to have to pull new Cat5 (or better) in order to wire those RJ-45's for proper Ethernet. Ethernet requires point-to-point connections, no daisy-chaining. No extra jack in the middle of a cable.

You may get away without pulling new wires, depending on what you want to achieve.

You can replace the current mixed sockets with 2 Ethernet sockets, each fully connected to one wire: one to the previous room and one to the next room. Then, in every place you're not intending to connect a computer, you link the 2 with a short patchcord. This way, you'll get a single "bus" running through your house. Wherever you want to tap into that bus, you replace the patchcord with 2 cords and a switch.

Depending on how many switches you'll need, it'll eventually be more expensive than simply rewiring entire house. However, a "tappable bus" is more elastic if you're not sure where the central hub should be, because all sockets are equal.

The only way you could make that work without pulling new wire would be to crimp those wires into proper ethernet and have a switch at each point in the line that takes the input and then has a downstream port continuing on the other wire in the box to the next jack. But that is really a terrible idea. I would only do that if I was only going to use it in maybe one or two spots. If you are lucky maybe it's cat 5e so you could get gigabit out of it. If that was the case the performance might be ok. Remembe that this is a hack. Not ideal at all.

You really need to pull new wire or consider a powerline adapter solution. That or just rely on wifi.