Well when the element went out it most likely would have taken the breaker out, I'm hoping this was the first thing you checked. Sometimes a breaker will trip and not look like it, usually we always physically flip the breaker full off, then back on.
After this you're going to need a voltage meter to test anything further. I'd start with the meter right at the outlet behind the oven its plugged into. I've seen shorted elements cause so much amp draw to occur that it literally burns one of the wires inside the outlet off the actual outlet. Your meter will be used to verify you are getting proper voltage at that outlet. Even if the microwave works, this oven is most likely 240V. That means you have 2 separate 120V lines a neutral and possibly a ground. If you lose just one of the 2 120V lines the displays and stuff may still work, but the full 240 V needed for the heating element will cause it to not heat.
If that tests ok, then I'd check next in the back of the oven where the power cord comes in. Usually there's a little cover there you can remove and make sure nothing burned off on the terminal strip, again with your meter verify you have the proper voltages there.
Likely labeled L1, L2, N, G
With your voltage meter probes on L1 and L2 you should get about 240 Volts AC (may be slightly more or less) Then from L1 or L2 to either Neutral or Ground you should be seeing about 120 Volts AC.
If all that tests out good you're then going to narrow it down to those two switches. The one that sets the temperature works by power coming in on one terminal, when it senses it needs to be heating it then lets that power coming in go out the other terminal. You can set the temperature up with the power on and check each terminal with one probe on your meter on one of them, then the other probe of your meter touching the unpainted metal somewhere on the oven itself. You should get a voltage reading on each terminal to ground (the metal), and read 0 volts if you touch one probe on each terminal.
That will tell you if the thermostat is good or bad, or at least working.
If the thermostat checks out then you'll need to test that other switch where it says Bake, Broil, or Off? Set it to Off, then with your meter test from ground to each terminal, one of them should give you a voltage reading. This one is power coming into the switch, make note of it. Now switch it to bake and test each terminal to ground, there should be at least two of them now you get a reading from. If not, then this switch is bad.
Does your broil still work? I'm guessing it does since you only mention replacing the bake element.
If so, and its that Bake, Broil, Off switch thats bad, you could get by till your remodel by swapping the wires on that switch so when it says Broil it actual bakes. Most people rarely use their Broil so this could likely get you buy without having to replace any parts on it at all. If thats where your problem is.
At this point though, there is nothing you'll really be able to just look at and determine. You're really going to need a voltage meter. Doesn't need to be an expensive one, just one capable of reading AC Voltage. Simpler the better if you don't know already how to use a meter. | 
