34 years old is, as you may have sensed, is an older-than-average age to enter consulting. However, your experience is an asset. 16 years in industry using formal frameworks and methodologies lends you client-credibility and marketability.
I would suggest as a next step you sell yourself as a specialist in a single, deep area. Are you an ITIL Service Manager? Or are you a PRINCE2 project manager? Or are you something else altogether?
The big IT consultancies have a need for all these specialist roles, and you should contact them - but only with a clear view of where your experience could best be used.