IL does NOT tax qualified withdrawals from out-of-state 529's. The only advantage IL gives to its 529's, that it does not extend to other states' 529's, is the ability to deduct your 529 contributions (up to $10,000 per taxpayer per year) from your IL taxable income.
Speaking of the IL deduction, you may want to rollover your LQ money into IL's Bright Start (or Bright Directions), thus earning an IL deduction (again, up to $10,000 individual or $20,000 for a married couple) for the BASIS portion only of the rollover. So this is your part (b) idea, but you DO get a deduction from IL. The only potential downside to this is that KS may want any tax savings back that you got when you contributed to their plan while you lived in KS--IF you earned any KS deduction, AND if KS "claws back" any such deduction should you close out your KS 529 for reasons other than covering qualified expenses.