Hellman: I took myself the liberty of editing your post to improve grammar/spelling/sentence structure for clarity, as I believe it's a great post, and it would be a shame if it was missed by other people just because it was somewhat hard to read.
I'd like to add that 345 grams of elemental bromine is equal to 111 mL (Br2 density: 3.11 g/mL), it is much more convenient to quickly measure out a certain volume of bromine (the bottle advantageously cooled as much as possible before its opened, but not below -5°C or it solidifies) than dribble with it on your expensive balance before adding it to the addition funnel. Bromina vapors are nasty, and remember to use rubber gloves - bromine woulds heal very slowly!
However, I have a few questions regarding the work-up:
After all the solution has been added, maintain the solution in the RBF at 65-70°C for another 10 min, your indicator here to know when to stop warming is when the solution becomes clear. Drive of all the MeNH2 by gently warming the RBF.
Is the part where you drive away the last of the MeNH2 gas included in the above 10 minutes, or is it 10 min plus some more, and if so, is that also done at 65-70°C?
the by-product being ammonium chloride. A nice boil with EtOH will remove all that, recrystallize, and dry the product in a desiccator over anhydrous CaCl2 to give Pure MeNH2.HCl.
Ammonium chloride is no by-product of the Hofmann reaction betwen acetamide/bromine/NaOH - if there indeed is any ammonium chloride in the crude MeNH2.HCl, then that must come from ammonia or ammonium acetate contamination of the acetamide prepared in the first step.
(https://www.thevespiary.org/rhodium/Rhodium/hive/hiveboard/picproxie_docs/000443020-file_dpki.gif)
Above is the reaction mechanism for the rearrangement pictured, and here are links to some references on the Hofmann reaction (http://themerckindex.cambridgesoft.com/TheMerckIndex/NameReactions/onr194.htm)
(http://themerckindex.cambridgesoft.com/TheMerckIndex/NameReactions/onr194.htm) and a small-scale preparation of Methylamine.HCl from Acetamide (http://www.rod.beavon.clara.net/hofmann.htm)
(http://www.rod.beavon.clara.net/hofmann.htm), similar to this one.
Also, the purification must probably be clarified for many people - you separate the unwanted ammonium chloride by taking advantage of the fact that MeNH2.HCl is soluble in in boiling ethanol (denatured, I assume, considering the cost of pure 95% EtOH?), while the ammonium chloride is not - the full procedure being described in Organic Synthesis Coll. Vol. 1, p 347 (http://www.orgsyn.org/orgsyn/prep.asp?prep=cv1p0347)
(http://www.orgsyn.org/orgsyn/prep.asp?prep=cv1p0347)
My last point is that your reference to Vogel's book is wrong, there is nothing about these reactions on page 246-248 in the 5th edition - are you thinking about the 3rd edition, or are your page numbers off?