I don't agree with the "via French" part. I think it was mostly via the Romans.
You can research this fact for yourself, but here's one example:
How French Has Influenced English
The English language has been shaped by a number of other languages over the centuries, and many English speakers know that Latin and German were two of the most important. What many people don't realize is how much the French language has influenced English.
Without going into too much detail, I want to give a little bit of background about the other languages which shaped English. It was born out of the dialects of three German tribes (Angles, Jutes, and Saxons) who settled in Britain in about 450 A.D. This group of dialects forms what linguists refer to as Anglo-Saxon, and at some point this language developed into what we know as Old English. This Germanic base was influenced in varying degrees by Celtic, Latin, and Scandinavian (Old Norse) - the languages spoken by invading armies.
http://french.about.com/library/bl-frenchinenglish.htm
And here's another more scholarly text on the subject, with references listed for further research. In any event, it is quite clear that neither Mexico, Spain, nor the Spanish language have had little to no effect on modern English, other than some new words and/or slang.
THE INFLUENCE OF FRENCH ON ENGLISH IN THE EARLY MODERN PERIOD
BACKGROUND
For a greater portion of the Middle English period (M.E.) French (Fr.) was the governing vernacular of England. It was the language of the ruling elite, many of which spoke little if any English, the language of the court, and the language in which polite literature was written. Hence, studies using the Oxford English Dictionary revealing that Fr. was the primary source of loans outnumbering Latin, the second largest source, four to one should hardly surprise us. As Baugh points out “where two languages exist side by side for a long time and the relations of the people speaking them are as intimate as they were in England, a considerable transference of words” is “inevitable.”
[snip]
http://www.chass.utoronto.ca/~cpercy/courses/6362Jurcic1.htm