Sparky has the right idea I think. If you don't mind the gate swinging in, you can cut the top and bottom 2x4s (or are those 1x4s?) off and add pressure treated lumber as framing sides, top, and middle brace. Add about 3 gate hinges on the side (might get away with 2, but 3 would be sturdier) and add a latch on the other side. This way you could use existing posts. Although, you may want to dig around the post closer to the house and add reinforcement to make it sturdier. Maybe add another brace to the corner to make sure the top stays toward the house more. You'll have to make sure the pieces you pad out with on the sides are the same depth/thickness as the posts so they line up properly.
You can add a rope or something to pull the latch from the opposite side to unlock it, or have it in a spot where you can reach over and pull it.
You can get something like this style (if they aren't too long)
I'm trying to find the kind I used... Not seeing them, but maybe something like this:
Not sure if they sell 3 hinge kits or not.
There's also the latch
Or
Now, you could make the gate smaller if you want and add another support post for the latch on the inside.
I suppose you could also figure out how to make the gate swing the opposite way and have a different type of latch and different style hinge.
Eddie, the wheel would be good over paving stones or slab, but not so great over dirt as it will sink and possibly snag on grass.