I do not agree with your last part: knowledge without experience is possible.
Yes but the judgment you do when you know the forms of pure intuition is different from an empirical judgment.
An empirical judgement is a synthetic a posteriori judgement, such as 'a rose is red'. It is directly derived from observation, but the predicate can not have universal value, therefore it cannot be used in science.
Then there are analytic a priori judgement, where the predicate is already implied in the definition of the subject, such as 'bodies are extended' or 'a triangle is a figure with three sides, ant the sum of its internal angles is 180°. This is an example where you don't need experience to gain knowledge - even though this kind of knowledge really is just an analysis of the working definition of a subject. The problem Kant has with this kind of judgement is that it does not progress toward anything, it does not add anything to the definition of the subject.
The main question of Kantian philosophy is whethere a third kind of judgement, the synthethic a priori judgement, is possible. This judgement should add something to what we know about the subject (synthetic) without being derived from experience (a priori). Those kind of judgement are the ones allowing the progress of science.
An example of this are, for him, mathematical laws. 7+5=12, in his opinion, is a possible sythetic a priori judgment, since the result 12 is not already implied in the definition of 7 and 5 and, moreover, the result of the operation is not drawn from experience. We don't need to sum 5 things to 7 things to know that the sum is 12: we just need a working concept of what a unit is, and even without experience we can get to the result.
Now, according to Kant, knowledge of the pure forms of intellect is a kind of analytic a priori judgement: it is the analytic part of the transcendental logic. It means that he is breaking down a fact (the phenomenon) into its constitutive parts. The pure forms of intellect are already implied in the phenomenon as its necessary formal principles.
Take, for instance, the category of unity: no phenomenon would appear if it was not conceived as a 'something', which is the same as to say, if it was not conceived as a 'unity'. Therefore, a formal principle organizing intuition into unities is one of the conditions of possibility of a phenomenon.
The point is: I don't really learn this from observation. It is not the perception of a phaenomenon X that gives me the knowledge that formal principles are necessary to have experience, but the fact itself of having experience already implies that. The fact that formal principles are necessary from observation is not learn through observation, but through analysis of the conditions of possibility of observation itself.
It is not like breaking a TV apart and watch inside. Kant never opens the TV: he analyses what kind of principles make a TV possible - doing this not require observation.