How could the Green Bay Packers let Kyle Orton and screen pass tear their defense apart? Well, Orton was pretty good against the Packers the last time he played against them (as a member of the Broncos) but this time they weren't able to force any interceptions. Also, the pass defense has been getting burned by running backs and tight ends all season long, so this game was really no exception.
The pass rush doesn't have much bite outside of LB Clay Matthews, and the only player to record a pressure against the Chiefs was S Morgan Burnett. The pass defense gives up short passes and crossing routes (to make the offense work slowly down the field) and the Chiefs took it. Arguably the defense succeeded because they forced the Chiefs to settle for so many field goals. Holding the Chiefs to under 19 points should have been good enough.
If they haven't been able to fix these problems so far this season, I don't know what they can do to fix it now. The only thing that could change is that they could play the safeties and linebackers closer to the line of scrimmage and take away those short passes. And that might be something they can do in their (hopefully) four remaining games this season at Lambeau because the cold weather will make it more difficult to complete deep passes. There isn't much to be done about the pass rush until they can get Matthews some help next season.