You’ll need to build an active amplifier to go from Mic level to Line level. Passive preamps do some clever things to match output and input impedance, but won’t actually make anything any louder (which is exactly what you’re trying to do). Difference between line and mic level here.
What you’ll need to do it build an amplifier which can amplify a signal between 100x and 1000x. Thankfully you shouldn’t need to provide much current, just more voltage (which makes the electronics easier).
You might want to check this out: http://www.instructables.com/id/Build-The-5-Microphone-Preamp/
What kind of microphone are you using? I’m assuming that it doesn’t need phantom power - which would mean a more complicated circuit.
That kit you posted isn’t really aimed at this kind of thing, but it might just work. It says it provides a gain of 100x, so could be at the low end. Worth a try and if nothing else it could be a good source of components for a second attempt. The thing you’ll need a keep an eye out for is that the microphone is providing a balanced signal (two wire both containing the signal but at opposite voltages), and the line output is going to be unbalanced (just one output wire). The lazy approach to this is to ignore the balanced signal and just use one of the signal wires - but you could get a noisier signal.
At the core of this, you’ll want to check out this Wikipedia page about op-amps. They’ll be your friend in this quest. https://en.wikipedia.org/wiki/Operational_amplifier