It seems that the original proof can be found in Volume 2 of "Gesammelte matematische Werke" [Complete mathematical works], published in 1930, which was compiled and edited by Fricke, Noether and Ore. More specifically, you can find the full proof -- two pages -- starting on Page 416, Section XLIV. This proof seems to contain the ideas of a suitable choice of $k$-basis with non-zero determinants.
Unfortunately, I am not aware of any existing English translations of this book, but it should be navigable if you are familiar enough in the area, as with all mathematics. The concluding line
... woraus folgt, daß alle $h=0$ sind.
should be recognisable in any proof of linear independence! Note that the full texts of all three volumes of this book are available online from the Göttingen State and University Library, at this link. The page you are interested in is this one.
