Let me get this straight :
You have a device that stop working, then you decided to troubleshoot it. Found that you were not getting a 'reading' off of the transformer with your equipment. I am not sure about that particular peice of equipment, is it a multimeter? If it is, it should be no problem to use its AC voltage setting and to measure the output off of the seconary of the transformer. You should be getting a reading of some sort when it is plugged it. Make sure that there is actually power going into the primary coil with the AC meter also. You decided to plug in random transformers to see if it worked? This is a bad idea without taking into consideration the requirements of the device.
The 'blown' transformer should indicate several important specs., more importantly the 'voltage output'. You need to take into account the configuration of the transformer (one secondary, center-tapped, three secondaries, etc.). You also need to consider the power capability, this is expressed as a current or a rating in Volt-Amperes.
Once you have these specs., you can find a similar replacement. You must be able to supply the same voltages as the damaged transformer. You need to make sure that the transformer has a power rating which is equal to or higher then the original. Also, the configuration must match.
If you have too much voltage, your circuit may not be able to handle the power losses associated or it may exceed voltage ratings. Also, if you don't have enough power, it may not be able to supply current during current demanding tasks (ie. highspeed upload/downloading, rebooting modem, etc.)
Hope this helps your understanding,
Steve