It seems there might be a discrepancy between what's transcribed as the ship online and which ship it actually was. I found the articles you mentioned about the SS Kaiser Wilhelm II, and those seem right. But looking at the manifests for the ship that did arrive on June 17, 1914 (or June 18? according to some pages on it), it seems the ship may have been the SS Kaiser Wilhelm der Grosse, also built by NDL. See, e.g., here: https://heritage.statueofliberty.org/show-manifest-big-image/czoxNzoidDcxNS0yMzM2MDIxOC5qcGciOw==/1. The affidavit lists the ship as SS Kaiser Wilhem d. Gr.
Dear Lindsey Chaffee,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located the series Passenger and Crew Lists of Vessels Arriving at New York, New York, 6/16/1897 - 7/3/1957 (Microfilm Publication T715) in the Records of the Immigration and Naturalization Service (Record Group 85) that may include the lists of the SS Kaiser Wilhelm II and, as a previous poster noted, the SS Kaiser Wilhelm der Große for 1914. You may be able to use these records to verify the arrival information (e.g. date and name of ship) of your great grandmother. Some of these records have been digitized and are available using the Catalog. For information about the non-digitized records, please email the National Archives at Washington, DC - Textual Reference (RDT1) at firstname.lastname@example.org.
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From the following article it looks like it went to Southampton for repair and then arrived in New York on August 6, 1914.
On 17 June 1914 Kaiser Wilhelm II collided with the 3,000-ton British steamer Incemore in thick fog off the Needles. Kaiser's hull was holed below the waterline, but the ship's watertight bulkheads held and the ship returned to Southampton under her own power. Kaiser Wilhelm II was west-bound when war with Britain began on 4 August 1914 and, after evading patrolling British cruisers, arrived at New York two days later.