I purchase a new water intake fitting for my karcher 395 and can not figure out how to swap it with the broken fitting

This will depend on where exactly it has broken on the inlet pipe and if Karcher sells the spare you need. Some inlet ports are 'screw-on' adapters and they can be swapped.

if your machine was factory fitted with 'total stop system' TSS and has always stopped in the past then it is an issue with many possible parts !

does the machinw still operate at full pressure ?
if the machine can produce full pressure then it only has to produce 20 % more to actuate the TSS piston. When the stop piston moves it presses against an electrical switch tjat breaks the circuit. Producing 20% more pressure requires that your machine has no water leaks in the pump, hose, trigger gun or lance. Once you release the trigger the path back from the trigger to the pump TSS piston must be fully sealed.....

If you have a Karcher then you should always post your machines barcode.
The karcher barcode is on a silver sticker and is usually 15 digits long. Example 156987400000123 is actually model 1.569-874.0 and serial number 00123....

Plastic Pump Manifold is cracked.
Can be replace at a local Karcher Service Center, or if you are mechanically handy. Replacement Manifold Complete, is available from www.karcher.com.
If it is under 1 year old, and under warranty Karcher will replace if you have a Receipt proof of purchase.

Is there a water leak from the bottom ?
Is so, then there maybe a small crack in the seam of the plastic manifold pump housing or one of the connecting fittings. A New Manifold Complete about $61.00. You may purchase direct from Karcher, and replace if you are mechanically inclined.
If under 1 year old, and you can find a receipt, Karcher will replace the whole Pressure washer under warranty.

Hi all you need is a 3/4 tap fitting from a diy store.
Maybe a hoselock 3/4 male and female fitting that will fit a standard 1/2" hosepipe. If struggling I can send you a link to the fitting on ebay but if you type in karcher tap connector you will see it by the seller vavcandwash.

Take a socket that is the same size as the black plastic and tap it down through the hole until it comes out. I have done several of these this way.

I found that by putting the housing in the freezer for a couple of hours, i was able to tap the metal and plastic body apart and install the new inlet elbow.

Had the same problem.Those hose inlet are very delicate.
I went to Home Depot,to the Garden Section in the hoses stand,and found a flexible connector for about $ 6.00 and I used it as a replacement for the broken hose inlet.
Good Luck!