I just took a look at that adapter you plugged into the Bose 1/4-inch input. That's a stereo headset adapter and not intended for what you are hoping to achieve. Essentially, it parallels a pair of TRS plugs and causes lots of problems.

Now, I do not have a schematic diagram of the Korg outputs, however, I'm going to assume that they are similar to the Yamaha outputs. If this is the case, when you are plugged into the mono output of the keyboard, a leaf-switch is enabled on the quarter inch output plug that combines both the right and left channels, thereby providing you with a clean, mono out directly from the keyboard. What you have done is taken a standard 1/4-inch phono output from each side of the keyboard, and plugged it into two TRS plugs that are wired together. Yes, you will get output, however, you will also get an impedance mismatch, which is not a good thing.

The solution is quite simple, though. There are three inputs on the Bose L1 Compact. Purchase (or make) a pair of 1/4-inch phono to RCA male plugs, and input your keyboard into the RCA plugs on the Bose. This provides a stereo channel input to the Bose coming directly from both channels of the keyboard. The output, of course, from the Bose, will be in mono, but it will sound very full and robust.

Hope this helps,

Gary