Will Change of Employer duration Affect ILR?

Dear Experts,

Thank you for all of you giving me continuous support through out my Tier 2 Journey.
I have a question below.

As my Previous employment end date 31/12/2016, I have received P45 from my previous employer but still not received 60 days letter from HO, do I need to submit my new application within 60 days from my last employment end date 31/12/2016 or can I wait for 60 days letter?

My solicitor advising me to submit the application before 60 days from the date of my last employment end date 31/12/2016; else if I apply after 60 days I won’t qualify for ILR in 5 years as my continuous lawful residence would be considered as broken.. Is that True? Please advice.

Thanks & Regards
J&J