Getting your foot in the door is the hardest part. The only advice I can offer you is to stick at it and keep building up and improving your portfolio (both your online one and your print one). If you want to link us some of your work, we can offer our advice.
Rather than asking for a placement out of the blue, try asking for half an hour to have someone look over your work and offer suggestions for improvements. Then, after your meet, ask if there's any chance of a placement. Offer to just come in for one day a week if needs be. Say you'll bring your laptop in and work quietly somewhere out of the way so they don't have to provide you with a desk. Just give them no excuse to say no (but don't be pushy).
If there's still no luck, just move on. At least you've got A) advice about how to improve your work B) a contact within the industry. Just try again later, ideally contacting the same person you spoke to directly. Remember to ask for their business card before you leave, and hand them yours (if you don't have cards, get some!). Consider asking for placements in other cities (or even other countries!). Studios love to have people with different backgrounds and cultures around.
If you're not on Twitter already, sign up right now and start following local designers and studios, comment on their blog posts or tweets and generally just engage with them. Go to local networking events or talks where the industry professionals will gather. I regularly see creatives from Manchester at events, and because I follow them on Twitter, I have an excuse to introduce myself. I've generally found designers are very friendly and willing to help students (don't forget, they've been through exactly what you're going through), so long as they're courteous and respectful.
Just keep at it. I've heard from many people over and over again that getting into the industry is often just a matter of time. Think of each challenge as a hurdle. Many people will fall at the first one, but if you keep leaping over them, the competition lessons, and eventually you'll reach the finish line.
