Hopefully you did a good job in splicing the extension wiring. I don't think that ou can have any kind of a permanent installation where both ends plug in. So, it would be like an extension cord.
I also don't know about what a good splice would be. I would imagine that properly sized wire nuts (Scothchlocks or equivalent) would work. To be really safe, you could make the connections inside of a metal electrical box and make sure that you ground the box. I'm not familiar with 220v dryer plugs, but newer designs should include a ground wire. You can put a plain metal cover on the box.
You should also use strain reliefs on both ends.
I have fixed more than one extension cord with wire nuts and electrical tape, but you have some heavy duty stuff here.
It would be easier to get 10 ga UF 3 wire plus ground cable to make the extension. Otherwise you have to keep the wires together somehow.
Depending on how your house is laid out, it might not be that difficult to just run a circuit from your fuse box. If you can use flexible cable rather than conduit, it could be fairly straight forward. You can find electrical code books at the library.