The problem you are trying to solve is to find values for \$\alpha\$ and \$\beta\$ that satisfy the two equations for \$K_P\$ and \$Q\$.
The problem you have solved is to find values for \$\alpha\$ and \$\beta\$ that satisfy the equation for \$\frac{|K_P|}{Q}\$, but not the separate equations for \$K_P\$ and \$Q\$. You need to add one of these in as a requirement.
I suggest you write these two equations out, eliminate one variable (\$\alpha\$ or \$\beta\$) and solve. If I have done it correctly, I think you will find a separate condition \$|K_P| > \sqrt{8} Q - 1\$, which your attempted design fails.
Also, your equations don't match your schematic - the capacitor values are swapped. This doesn't really matter as you assume they are equal, but it's still wrong.