Lithuanian does not use double letters like "-ss-", so assuming that she was ethnic Lithuanian (and name alone will not be determinative), it would be spelled something like Kasakitas. The ending is Lithuanian, though such an ending is also used by Greeks.
The nearest spelling I could find in the online phone book for Lithuania was the masculine name Kasakaitis (pronounced kah-sah-KAI-tihs).
Now ethnic Lithuanian women used different endings to masculine names depending on their marital status. A woman married to a man named Kasakaitis would be Mrs. Kasakaitiene. Their unmarried daugther (i.e., maiden name) would Miss. Kasakaityte (though prior to the standardarization of Lithuanian after WWI, it might well have been Kasakaitute.)
When immigrants came into this country they had complete control over their own names, changing it at will, though often doing so to conform to their English speaking employers, neighbors and merchants. Immigration officials, contrary to popular belief did not change anyone's name. Their duty as was the duty of the shipping company was to write the name as carefully as they could to correspond to what the immigrant told them their name was. Since most immigrants were illiterate and could not write, all documents had to rely on the pronunciation of the name as made by the immigrant. Thus many such names were written phonetically, often using the letters of the language of the clerk, e.g., if a Lithuanian used the letter "s^" in their name, pronounced "sh" as in the English "shirt", but the shipping clerk was Polish he might well have used the Polish letters for that sound "sz".
The main issue for the shipping company as well as immigration offiials was to document the name as presented by the immigrant. Serious discrepancies could lead to the immigrant's being deported upon arrival at the cost of the return trip charged to the shipping company. So the incentives were powerful for all to get the names as close to the immigrants preference as possible. The only time a U.S. immigrant official changed a name was when the immigrant claimed it was mis-written on the ship's manifest, in which case, sometimes the immigration official would write a correction over the name on the manifest. But this was quite uncommon.
At the time immigrants freely changed the spelling of own names, sometimes several times, sometimes to an extent that it no longer resembled their original names. Mostly, they dropped the use of different endings which Lithuanian used extensively throughout their language and shortened their names, often using phonetic spelling for Lithuanian letters. Thus a man named Kazimieras in Lithuanian changed his name perhaps to Casimir or more often to Charles. Ona became Anna. Marjiona became Mary Anne or simply Mary or even simply Ann. Same with surnames. My paternal grandfather was Petrauskas. He used the Polish spelling of his name when he emigrated in 1913 to the Polish spelling Petrowski, but when his son, my father, arrived in 1922, he almost immediately changed the name back to the Lithuanian Petrauskas, then very shortly after that while in school, he changed it to Peters. My maternal grandfather changed his Lithuanian name, Kuckailis (pronounced koots-KAI-lihs) by dropping the ending "-is" and converting the unmarked Lithuanian "c" to its phonetic equivalent, "ts" as in "bits" which is how the plain "c" is pronounced in Lithuanian, so he ended up using Kutskel. Oddly one of his chidlren later changed Kutskel to Kutsel and his offspring to this day are the only descendants to use Kutsel rather than the Kutskel everyone else used. So you see, none of these changes were made by immigration or shipping officials but all were made by the immigrants or their families at will.
The law did not require anyone to "register" a new name. But immigrants discovered that it was important to be consistent when making records, e.g., draft registrations, social security applications, naturalization records, so they often tried to keep a name constant after they changed it lest officials become confused or suspicious about the indenity of the person. But it was the immigrant who had complete control over changing his name, no court requirements, no officials requiring it, etc. Now some employers may have forced the issue early in an immigrant's career before the immigrant had time to settle on a spelling, and that might have been an external source for a name change that the immigrant accepted, or perhaps facing arrest, he may have tried to avoid prosecution by using a different name than he had peferred, but even so ultimately he had complete and exclusive control over the spelling of his name.