Here is what you should do temporarily until you properly open the wall and lower the drain pipe.
- The trap is in the wrong place. There is an open air path from the drain pipe to the drain basket and laundry. Hence the smell. The larger sink should use a 90 degree bend to join the tail of the drain basket BEFORE the trap. This picture shows the overall layout, it does not show the correct fittings to use:

Same design, presented on the original photo.

- The laundry should not be joined with electrical tape. Usually appliance hoses are connected to such fittings with hose clamps. If that grey pipe is rubber, that is what you should do.
The above isn't a good solution but it's a lot better than what you have. The main sink may not drain well. The laundry may wash up into the main sink.
Or
There may not be enough vertical space to configure this. I can't tell from the photo. If not, you'll need a pump. Move the trap as described above but join the main sink to it via a Utility Sink Pump that would sit in the cabinet. You'll need electricity there. The cost of doing this may be similar to fixing drain connection properly.
Or
Get a shallower sink and eliminate the drain basket. Then you'll have a vanilla installation.