I would assume that this question should have the obvious answer, that since $\frac{1}{0}$ is undefined, this expression is undefined also.
However, given that we can solve arithmetical problems with arithmetically undefined quantities (as far as the real numbers are concerned) such as $i$ on the premise that it eventually cancels out, why can't we do something similar by simplifying the expression as $\frac{1}{\not0}\cdot\frac{\not0}{1}=1$?
$\frac 10$ is not like $i$. When you adjoin a square root of $-1$ to the reals, you get something bigger, namely the complex numbers. You can define a multiplicative inverse of $0$, but it's not very interesting. Let's see what happens.
Let's define $a$ to be the multiplicative inverse of $0$. Then $0a = 0 = 1$. For any element $r$ of this new algebraic structure, $r=1r=0r=0$. Hence it only has one element, which you can call either $1$ or $0$ as you choose. In this new structure, $$\frac{1}0\frac{0}1=1$$ as you wanted, but we've lost everything to achieve this. It's not worth it.