Bombelli did not have a general method for such transformation, nor does it exist. Doing so is equivalent to solving the irreducible case of the cubic, which is impossible in (real) radicals. It is solvable in inverse trigonometric functions, as Vieta showed later, but simplifying his trigonometric expression is itself a chore.
However, when everything is an integer one can find the solutions by educated guessing. In this case, according to Katz's History of Mathematics, 12.3.2, Bombelli used indeterminates $a,b$ to set up (in modern notation)
$(a+\sqrt{-b})^3=2+11\sqrt{-1}$. Given his rules for $\sqrt{-1}$, this quickly leads to $(a-\sqrt{-b})^3=2-11\sqrt{-1}$, and then to
$$a^2+b=5;\ \ a(a^2-3b)=2.$$
By simple divisibility considerations, there are only four choices for $a$, and Bombelli shows that $a=2$ is the only viable one, with $b=1$. Note that attempting to solve this "honestly" leads to another cubic $4a^3=15a+2$, no better than the original one.
LaNave-Mazur's Reading Bombelli gives more insight into the way Bombelli handled cubics, and complex numbers that came up in them. As they note, Bombelli's "procedure" was generally not very tidy:
"if you open
Bombelli's treatise you discover nothing resembling complex
numbers until page 133, at which point certain mathematical
objects (that might be regarded by a modern as
"complex numbers") burst onto the scene, in hill battle array,
in the middle of an on-going discussion. Here is how
Bombelli introduces these mathematical objects. He writes,
"I have found another sort of cubic radical which behaves
in a very different way from the others"... Complex numbers, when they occur in Gerolamo Cardano's earlier treatise Ars Magna, occur neatly as quantities
like $2 + \sqrt{-15}$. But they appear initially in Bombelli's
treatise as cubic radicals of the type of quantities discussed
by Cardano; a somewhat complicated way for them to arise
in a treatise that is thought of as an organized exposition
of the formal properties of complex numbers!"