The rule and the norm in the Polish language is generally to decline whatever can be declined. Thus, the surname Kołodko should be declined, and indeed, it often is: I often heard "ministrowi Kołodce". Yet, some people, including politicians, insist - against the language norm - that their name shoud remain undeclinable. That was, for example, the case of the former PiS minister Zbigniew Ziobro, a case widely known in the Polish media world. He wanted TV and press journalist to leave his surname undeclined, yet the linguist who was telling the story to us at a language training at work, said the former minister was not right demanding this. Some surnames, however, may be left undeclined by tradition, but I am not able to give you any examples right now.
As for the name Kościuszko, I think the fixed historic tradition is to decline the surname, so you should always hear: Naród zaufał Tadeuszowi Kościuszce.
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