OK, dump the solder. 50/50 is plumbing solder. I hope its solid. If it has a flux core, and you have gotten some of the flux on the wire, cut that section off NOW.
Its acid core, and it will eat up the connection, and then slowly eat more of the wire under the insulation.
You say its a Weller pencil. Does it have the sleeve that unscrews down next to the handle and slides off the barrel?
If so, unscrew it, slide it off, pull the tip out, and see if it says 7 on the end.
If it does, then make sure its clean. Not shiny, just no crud on it, and put it back in.
DO NOT grind, file, or sand the tip of the tip.
Get some good 60/40 rosin core solder. Try Radio Shack. They will probably only have lead free, but it will work if its rosin core.
Wrap a small amount of the solder around the end of the tip, hold it over something wet, or that will not mind having hot solder dripped on it.
IOW, not the cat......
Plug in the iron, and wait for the smoke to clear.
Wipe where the solder melted off with a wet paper tower. Just a quick swipe.
It should be all shiny. If not, try melting some more solder on the dark spots to see if you can get the solder to stick to the end of the tip.
If you used the acid core with that tip, you really should replace it, but "wash" it a lot by melting a lot of solder on it and letting it drip off.
Then wipe it with the wet rag again.
When its nice and shiny, touch the solder to it to leave a very small blob.
Press that blob to the wire you are going to solder. Then briefly touch the solder to the point where the tip is touching the wire.
The wire must get hot enough to melt the solder its self, but by putting a tiny amount between it and the tip, you create a better heat transfer connection which lets the iron heat the wire up faster.
Then, when you touch the solder to the wire, it should melt. Watch it closely. If it just beads up, the wire has something on it that is preventing the solder to combine with it, or its not quite hot enough.
When its right, the solder will just soak into the wire and become part of it.
If you iron does not have the sleeve and the tip with the 7, but just has a thick wire looking thing that screws into the end of the element, unscrew it, clean the threads and tap any flakes out of the hole in the element.
If the end of the tip is crusty looking, take a file and sharpen it to a chisel point until you see clean copper.
Then screw it back in, wrap some solder around the copper part, and do like I said before.
Remember, if you have the tip that just sits in the end of the element, held in by the sleeve and a magnet, DO NOT file it. Its copper plated iron, and if you file off the copper, its a pain to make work again.
The screw in is solid copper and can be filed.
Hopefully you have the 7 one.
Here is why.
Most pencil irons take forever to heat up and cool down.
Not those Wellers.
With a regular iron, the heating element is sized so that when it gets up to about the operating temp, the heat output matches how fast the heat radiates away, so it holds at that temp.
But, that means that it heats up slowly.
The good Wellers have an oversize heating element that would burn up whatever you tried to solder.
But that tip has a special material on the end that goes into the element. It has a Curie temperature of the number stamped on the end times 100.
That 7 mean 700 degrees.
There is a magnet in the heating element hooked to a set of contacts. When you put the tip in, the magnet gets pulled to the end of it against a spring, and closes the contacts.
That turns on the element. Which heats up fast.
When the tip reaches 700 degrees, the material at the end becomes nonmagnetic, and the contacts open. When it cools down, it becomes magnetic again, and turns the element back on.
Thus, they use a big element that can crank out the heat, and heat up fast but its temp regulated.
They make different temp tips, but 700 is the most common, and works well.
But if yours has another number, like 6 or 8, its still the good one.