You could easily use a digital controller. I would buy a PID from http://www.auberins.com/. either the SYL-1512A for $37 or if you want to ramp and soak the Model: SYL-2352P and if your running (hopefully) 220v get 2 MGR-1D4840 40 amp SSRs. Yes you can get away with the 25 amp models and save $4 apiece but, a little heavy is good for dependability and long life. Then you need a thermocouple Model: TC-K-KLN is a type K that come with a ceramic mounting block, wire and ceramic "beads" to help protect it.
You have 2 options
1. rip out the present setup on the kiln and wire in the new set up using the diagram below to connect the power and the elements or
2 mount your new PID, SSRs and thermocouple in a box with a 220v cord and plug and a 220v femal plug that will accept the Kilns plug and just plug the kiln in and turn its current setting to high and let your new PID controlled setup run it via having your thermocouple wire exit the new box and insert into the kiln. This setup has many advantages as it can be used for other applications. You could even wire a separate 110v outlet off one of the SSRs and ground and use it to run a 110v device like a toaster oven for tempering or a PID controlled forge.
You do not need the LED that tells you the power is on. The PID being on tells you that. The one telling you the coils are on is nice, but not necessary. If you make the controller as a separate unit leave off the switch that shuts the coils down when you open the lid unless you want a separate set of wires coming from the box that plug into a switch on the door of the kiln.
To wire a 110 outlet you would need to install an extra wire off the output of 1 SSR and get the other 110 leg off the 220v ground wire not shown in the drawing. You would still plug it into 220 to power the control box although you could install a 110v cord and switches to allow it to work on either. The PID itself doesn't care if it has 110 or 220 to power it. Put you need 220v to run a 220 device.
With a kiln it works best to have a hole in the lid to hang the blades inside on a wire and have a piece of kaowool or better yet 1/2 a soft fire brick to cover the hole. Also because heat rises the top portion of the kiln will be somewhat hotter than the bottom!