You need to visit the big storage rooms in the second half of that entrance mine. IIRC, they're the only big separate rooms in that section of the game.
You'll find a pickaxe there, some string/cord and you need to knock down a locked door you come across (use the pickaxe, swing at the door and hit it until it flies off its hinges). Inside, there's a smaller barrel with black gunpowder that you also need to break apart with the hammer or the pickaxe.
In the office room of the mine (with the typewriter, in the first half of the same mine), you'll find Baxtrin in a grey jar. It's highly probable you've already taken the Baxtrin by that point, so have a look at your inventory. You need to douse the string in this chemical (just combine it in the inventory) and then you need to apply the string on the gunpowder pile from the barrel you broke open.
This will give you a finished piece of ignition string/cord, which you simply take to the TNT barrel, attach it, then light it with the lighter. Be sure to run away a bit and hide behind the corner while the explosion goes off ! You wouldn't want to be caught in the TNT exploding, obviously.
P.S. The storage section contains a working electric fence that you need to disable. Just put some planks or boxes over it, cross over, disable a battery that's supplying it, and continue. IIRC, the locked room behind the electric fence is the one where the gunpowder barrel is stored.
P.P.S. The room with the gunpowder barrel for the ignition string also has 2 or 3 sticks of dynamite lying around. If you want to, you can take them with you and throw them as improvised explosives. They're loud and a very rare item, though, so use them wisely.
"You... silly Billy !" (Clarence, Penumbra : Black Plague)