Cedar trails.
Ceder Trails is your best option, it is a really cool area with boardwalks. We walked there with my father in law who was in a wheel chair.
For a GREAT summary of the length of each trail within Glacier National Park, along with details, I'd consult this website (specifically devoted to hiking trails within Glacier NP) first: http://www.hikinginglacier.com/glacier-national-park-trails.htm
Chelle Coffee, I suggest you call Glacier and ask a park ranger your question. A lot of the trails are uneven and probably wouldn't be good for someone with a cane. It's a good bet that going to the sun road will still be closed in May or early June due to snow. It usually is. It's still beautiful though.
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