Section 8 rent increase San Diego

[Sam K.]

1. I recently purchased a duplex, one unit is Section 8 and has the same tenant for the last 15 years is what the selling agent told me. Its in the zip code 92117. Outdated 1b/1b.

1. The contract submitted to city SDHC is $1350 per month as rent by previous owner.
2. The last contract signed with tenants in 2018 is $1450 per month (but SDHC does not have it). This was done by previous owner. Possibly because the son moved in to take care of mom (~82 years old), Only the mom is on the lease. I do not want to disrupt their living situation since the tenant (mother is very old and has health issues). Its a month to month and still under the previous owner. I haven't created the lease under my name. I will pretty soon. 
3. I want to increase the rent to between $1595 up to $1650, which seems to fair value for this zip code IMO. Owner pays water.
4. There is no security deposit on file, I want to collect one months rent as security deposit too.

Question: Is there a limit by what % I can increase the rent? Should I bump the rent first, get that sorted and then create a new lease? or the both together? Since once I create a new lease the minimum term is 6 months and I can't change the rent once lease is in place. Any other things that I need to be careful about?