That's a tricky question, because a lot of problems can be circumvented by saying 'because magic'.
First problem is that ice is much more brittle then steel, so a cutting sword made of ice with be shattered easily. But it seems that our protagonist can prevent it by magic.
Second problem is the density. The average density of steel is around 8 gram/cm3. The density of ice is 0.9 gram/cm3, which is almost 10 times lighter. So if you go just by replicating the shape of a katana blade in ice, it will weigh around 90-110 grams (katana blade without hilt and fittings is supposed to weight from 800 to 1000 grams, as far as I remember).
Using katana hilt with the blade as light as 100 grams will be a problem of its own, if you want to use it as a sword, especially against other people with swords. A weapon that light will have problems cutting because of the lack of the mass in the blade, it's easy to block or batter aside with a heavier weapon. Generally, cutting swords rarely went down below 600 grams in overall weight, and specialized cutting swords (ones able to do damage to bones and wood) very rarely weighed less then 800 grams (Here is some data on the weights of different variants of British 1796 light cavalry sabers, one of the lightest effective cutters I know of). But calculating the weight of a cutting sword mathematically is a very tricky proposition - it's more of an engineering task, not purely mathematical. A lot of nuance lays in the physical properties of materials, their flexibility and strength. If we assume a material with infinite tensile strength, infinite sharpness and infinite slickness (zero friction), then we may go lighter then 600 grams for a sword.
As far as I see, your character can circumvent the problem in two ways, both of which will be decidedly magical. First, we can assume she can also vary the density of ice at will - then your ice blade would weight just as much as a steel one, and for all effects and purposes it will behave as a metal weapon, as far as handling and balance is concerned. So upwards from 800 grams if it's a katana blade, less (around 600-700) if it's a thinner and broader one, like tulwar or 1796.
Second variant is treating 'water' as an advanced nanomaterial. Here, again, we don't care that it's water before magic starts acting on it. In effect, what you want is a mono-molecular blade - one molecule in thickness, but of the length and breadth of a sword blade. The amount of water for this task will be trivial, less then 10 grams, I think.