How do I get beyond 60% or 70% at the chair level in the "Quest" part of the game?
"Quest" mode has 5 levels. Each level has 8 puzzles. You have to complete at least 4 puzzle in a specific level to be able to proceed to the next level. The percentage on the puzzles screen tells you what percent of the total puzzles (namely 5*8 = 40 puzzles) you have completed. So, each completed puzzle gives you 2.5% progress.
That being said, let me answer your question. If you have completed all puzzles till you get to the last level (the chair), then you must have got 80% progress. If you are below 80%, since each completed puzzle gives you 2.5% progress, you can select the uncompleted puzzles in the previous levels to increase your percentage.
In short, in order to get beyond 60% or 70% at the chair level, you need to complete enough number of puzzles no matter what kind of puzzles they are or to what level they belong.