I have an HTD Ultrasonic Cleaner model 113htd. All the bells and whistles. Around $55-$60 at Walmart or on Ebay. Basket, lid, heater, stainless, 1.3 liter, good timer (1-99 minutes), really good deal.
If you can't justify an ultrasonic cleaner then I highly suggest lemon juice. Remove all the o-rings and clean them in mild soapy water. Soak all the hardware in lemon juice for 15-30 minutes depending on how dirty it is. You can pick it up out of the lemon juice to check if your hardware is clean or not. Rinse the lemon juice off with hot water and wipe the water off as best as you can then set it aside to finish drying. Does wonders on brass, copper and stainless. Keep in mind nothing brings back that brand new shine except for actually polishing. You can use hot sauce too but it's a good bit messier and lemon juice is cheaper. You can even store the lemon juice in a plastic container in the refrigerator and use it again later. You'll know when to replace the lemon juice.