You'll not get far in the mines before running into your nephew, who tells you that the key needed to proceed through a gate here is located in Transylvania.
The trip to Transylvania is a brief one. From the start of the Transylvania area (where you'll appear automatically), head right along the ground floor, until you come to the suit of armor near a mirror. Bounce from the area of the floor slightly left of the suit of armor, up to the mirror, and you'll find the room that contains the key (or if you prefer, you may collect all sorts of goodies before returning to the mirror to grab the key).
With the key to the mines in hand, you can exit through the room's left mirror. Now you'll be back to the level select screen, and you should go ahead and select the mines again. Now you can continue past the gate where your nephew waited.

You'll find a circular block, which you should strike so that it flies to the upper right and knocks down a bat hanging from the ceiling. Hop along the ledge to the right, and in the air between it and the coal box ahead. You'll cause two treasure boxes to appear, which allow you to easily reach the one on the ledge to your upper right. That box contains a large red diamond worth $50,000, so collect it first. Then grab the contents of the other ones; they both contain standard diamonds.
On the way right, swing your cane at the coal box to reveal a large diamond, then hop right from there toward the chain to your right. Strike the circular block there so it takes out the bat to your right, then keep moving toward the right to another set of circular blocks. These won't be able to strike the bat, but you can defeat it with your cane and move onward to find both an invincibility medallion, and two treasure chests that contain an ice cream cone and another large, red diamond. The coal box here also hold a large diamond.

With those items collected, head back left to the chain and descend it. In this room, you get to choose from several directions. Begin by heading to the right, until you're standing under a ledge where Mrs. Beasley will toss down some tasty treats. There's a wide pit to the right of that. Jump out over the pit with your pogo aimed down, and you'll be able to pogo off the backs of enemies that leap from the pit and head along to the right, where you'll find a coal chest that holds a large diamond. Jump over that next pit with a standard leap and you'll find a treasure chest with a hidden treasure worth $1,000,000.
From the chest, head right to a chain, and climb down into the next room. The coal box you find here contains a $10,000 diamond. Grab that, and collect the cake from the treasure box to the right. You can descend the next chain, too, and collect diamonds from the recesses in the wall along the chain while you descend (one of them contains a large red diamond worth $50,000).

There's no way back to the start of the stage from here, so you can either die on purpose to reappear at the start (which is what I do; there's a 1-up icon nearby, I swear) or you can go on to face the rat. For now, I'll assume that you die on purpose, then make your way back to the room just left of Mrs. Beasley.
This time, instead of heading right, drop left and hit the coal box for a large diamond. Then walk through the wall to its left. On the other side of that wall, you'll find a large treasure chest with a 1-up icon. Move left again and you'll find a coal box left of a chain. Hit the coal box for a large diamond, then hop on top of it to make a large, red diamond appear.

Once you've collected that, climb down the chain into the next room below. You'll land on a treasure chest. Hit that with your cane to cause another large diamond to appear. From there, jump left over the top of the piranha plant. You'll make treasure chests appear, and you can bounce over these to collect treasure in the next room, located just over the ceiling.
When you've done that, return to the previous room and go ahead and climb down the chain. You're now in a series of catacombs. Head right and hop over the series of circular gray blocks. There's a pit of water ahead, and a green monster will leap from it as you cross. Make sure your jump either clears him, or that you pogo off his head. The next pit is the same case, and the treasure chest on the next ledge contains an ice cream cone. Pogo off the enemy to the right to make it to the next ledge, and from there keep crossing over pits while avoiding bats and the sea monsters. When you reach the right side, Launchpad will offer you a ride back to Duckburg. If your score is right, you might as well accept it.
Once you're back in the mines, you've really done everything you need to except defeat the boss. With that being the case, go easy on yourself and cross over the pit to the right of Mrs. Beasley as a shortcut to the area boss. From where you found the one million dollar treasure before, continue right, climb down the chain, then head left to the boss room.
Marble King
As bosses in this game go, this fellow's actually pretty easy. Probably because he's so elderly. But then again, so is Scrooge.
Basically, he rolls into a ball, then will roll around the room in either a clockwise or counter-clockwise pattern. When he reaches the bottom center, he will pause long enough for you to pogo him, then start rolling again. There's really no way to predict which direction he will roll, so stand on the ledge just to his left or right and jump outward as he rolls into the ball so that he either passes underneath you, or rolls to the opposite side.
Either way, he shouldn't be able to hit you. A few pogo stomps to the head will send this fellow rolling to the time-out box, and you'll be able to collect the mine's treasure as your reward.
NEXT: Transylvania