Networking and Asking for Internships

I'm trying to move into Consulting, with an emphasis within Public Sector initially.

I work at a small consulting firm now and I am due to resign in the hopes of securing a few more internships before September.

I was wondering what is the best method to approach firms for internships when they have not advertised anything. Unpaid work is cool with me providing its relevant. I had a few concerns:

1) Would you think its wise to mention in an introductory email asking for the position about beginning my masters in September?

2) Whats the best way to approach them? I largely have been saying something like: 'I am a law graduate at xxxx due to begin a Masters in September with hopes of continuing within consultancy. I am enquiring as to whether I would be able to gain some experience via an unpaid internship or formal arrangement at xxxx over the course of this summer. As I mentioned I am more than happy to work in an unpaid capacity. I would really value any potential positions you may have available' I haven't heard back from a few while another mentioned my experience isn't what they are looking for. So got me worried my approach may suck.

Any help guys?