How do I connect both my dishwasher and washing machine waste pipe to the same sink?
The part you have is for connecting to the top of a standpipe trap. Not for adapting a existing single-spigot P trap. The hole at the top is for air admittance I think.


  What part should I buy?


I would buy a "Washing Machine Trap with Double Spigot"



Example: McAlpine WM11 £10 ish.

Since your waste connection is quite high this might mean adjusting the other plumbing.