A.D., with Jim staying up until the wee hours of the morning to help you, the least you could have done is stay up with him!
I don't know nearly as much about espresso machines as Jim does, but I do know a bit about electronics and the loud pop you heard suggests that the failed device is something with sufficient mass to make a loud pop when it shorts. The heating element is the #1 guess, and Jim has already told you how to check that.
You can buy for as little as $10-$15 a digital multimeter, or borrow one from a neighbour. One of its functions will be for testing resistance, and will be indicated on the function selector switch with the sign for "ohm"... it's kind of an arch with feet... look it up on the internet.
UNPLUG YOUR MACHINE BEFORE TESTING
With the meter turned on, if you touch the two probes together, the meter will show 0 ohms (a "short", no resistance). If you then do what Jim suggested - remove the wires going to your boiler's heater, and hold a probe against each of the two connectors on the boiler (doesn't matter which is which) then the meter will give you a basic report on it's health. 1. If it shows some resistance - Jim says it should show about 10-15 ohms - then it's probably OK, you have to look elsewhere. 2. If it shows 0 ohms, just like when you held the two probes together, then the element is shorted out and you've found at least part of your problem. 3. If it shows the same as it does when the probes aren't contacting anything, then it has infinite resistance and has blown itself apart. Based on your "pop" description, the bets would be on #3.
If the element tests OK, then check the thermal fuse the same way, making sure to remove the wires first. This time, you WANT to see #2, 0 ohms, a short, showing that electricity can pass right through it.
You can work through parts of your machine like this, checking the path for the electrical energy all the way from the plug, and possibly find what's wrong yourself. Then, if you don't want to make the repair yourself, you'll at least have a leg up when you take it to the repair place.
Good luck.