I understand your allowing that Congress has a function of impeachment, but is eligibilty a high crime? I believe that the SCOTUS must be able to rule on Obama's eligibility for the office. Then it will be up to the Congress to address the matter if Obama is found ineligible.
The Supreme Court has already tossed aside this stuff before....
http://latimesblogs.latimes.com/washington/2008/12/obama-birth-1.html
The justices do not deem it worthy.