Sorry.It seems it is not your Anastasia GLADKY in the SSDI. The one that is, she passed away in 1980.
22 GLADKYs passed thru Ellis Island. Each language has its own alphabet and pronouncation.Just an example :The Slovak alphabet has no W and Polish has no V.It is possible that Gladky is Russian spelling and Gladki Polish. In Slovak, the surname Gladky would be Hladky. To me the first name Anastasia and Nicholas would suggest Russian, Greek Catholic or Orthodox belief.That is not to say that there is no exception to it. The SKI ending suggests Polish roots and SKY Slovak roots. Need that precise name of a town.