I am going to assume k' is a pseudo first order rate constant:
First ln[A]t=ln[A]0 - k't
Plot ln[A]t vs. t
This will give a negative slope with a gradient of -k'
Then using this value for the rxn rate (assuming this is a first order rxn):
ln k' = ln A - Ea/R * 1/T
Plot ln k' vs 1/T to get another negative sloped graph, with -Ea/R as the gradient.
From that you get Ea.
Look up Enolates for the mechanism:
The acidic H causes enolisation (the formation of a dbl bond adjacent to a hydroxyl carbon) C=C(OH).
The enolisation is highly in favour of the ketone so the enolate is the rate determining step/species of the reaction.
Although in a very low conc. (approx to the order of 1 in a million) it is a very reactive species: The iodine will add to the enol:
The OH group has a lone pair, which falls into the dbl bond, which then attacks one of the Iodine atoms of I2.
The resultant compound has a C=O+H group, which naturally falls to form a keto group again + H+.
Hope this helped.
An attempt at a mechanism (RDS=rate determining step):
C-C-C=O: H+H2SO4 < - >R.D.S C-C=C-O:H I-I (electrons from O go to dbl bond, electrons from dbl bond attack an I) -> C-C(Br)-C=O+H -> Product