Turn signal switch is broken ?

It is unlikely that the indicator stick for an MX5 NC Mazda will fit a Mazda 6 GY 2005 model. While both vehicles are produced by Mazda, they are different models with different specifications and parts.

This is the one I have bought.

This is the one that's broken :(

This is called a "Combination Switch" here is how to replace it in the images below. Check out the images (below). Please let us know if you need anything else to get the problem fixed. It doesn't look like you have the correct part.