I'm not overly familiar with the (h,k,l) calculations. I'm actually just learning them myself but everything looks good to me. You however have made a math error:

[itex]\frac1{{d_{hkl}}^2} =\frac{h^2}{a^2}+\frac{k^2}{b^2}+\frac{l^2}{c^2}[/itex] [itex](d_{hkl})^2≠\frac{a^2}{h^2}+\frac{b^2}{k^2}+\frac{c^2}{l^2}[/itex]You can't simply invert both sides of the equation like this with addition of fractions. It'd be like saying:

[itex]\frac11=\frac13+\frac13+\frac13[/itex] [itex]\frac11≠\frac31+\frac31+\frac31[/itex]If you correct this error you'll get 5.30Å in agreement with your friend. Hope that helped but given the delay I suspect you've already figured the issues out by now.