I don't think there is any quick way to do it unless there is some sort of family emergency provision I'm unfamiliar with (perhaps adoption, but that can be long-winded have a read of
this, I suspect it won't qualify as there is already a family relationship there). If you were a US citizen you could sponsor your brother for immigration but there is a ten-year wait for that category (sibling sponsorship).
I think your best bet might be F1 student:
http://travel.state.gov/visa/temp/types/types_1268.html
It can be a little dicey with children under 18 to get an I-20, but it is possible. Note the restriction for attending public schools: maximum 12 months, so at a bare minimum you would have to figure out how to pay for two years of private school, (assuming you can get an I-20 from a public school anyway, which is no mean feat, I suspect).
Note also when you read the regulations that the regulations for Canadian citizens are somewhat different than other nationalities, i.e. less restrictive, as Canadians do not need a visa, but they do need the school to sign off on the I-20.
Check with the US embassy/consulate in Canada and have a poke around the State Dept. and USCIS websites for more information.
There's also an obscure IRS rule that isn't mentioned anywhere on there, essentially this means that funds from outside the US (e.g. from your brother) given towards the student education (including living expenses) are subject to US income tax, unless a 1040NR is filed by that individual plus a Form 8843 every year. In addition, that exemption only holds for five years (total, regardless of how many different schools she goes to), so for example if your niece went to college after school in the US it could get awkward to fund it.
Steve.