Look behind your speaker. You will see two metal contacts.
Cut a length of speaker wire. This can be as long as necessary.
Strip 1/4 inch of rubber protective coating from each wire. You will remove coating from four wires altogether.
Melt some solder on the tip of your soldering iron. A small bead will do.
Touch one of your wires to a contact behind a speaker. Transfer the bead of solder to this junction. Tug gently to ensure a connection.
Touch your remaining wire to the second contact. Melt more solder and attach this wire.
Unscrew the body of your 1/4-inch male adapter. This will reveal two contacts.
Solder your two remaining wires to the two contacts on your 1/4-inch male adapter. There is no specific order.
Plug your new microphone into a recording device and test it by speaking into the center of the speaker.