Wiring in older homes can be very hard to decipher if you are not a professional electrician. First of all any electrical installation should be done by a qualified licensed electrician. Dim lights are usually due to a loose or non existent Neutral wire (The White wire). In older homes it was common to borrow a neutral wire from any circuit that was close to where the electrician needs it. This leads to overloading of the wire and can lead to a fire. It was also common to switch the neutral instead of the hot (Black) wire and this leaves power to the bathroom vanity fixture all the time with no place to go but to a fault or ground. If the neutral is not connected to a good path back to the electrical panel then the power may be flowing back to through a ground of some sort that may not be a solid connection leading to an unsafe situation. Finding the loose connection will most likely require an electrician to find.
