In this instance, assuming you bought/sold 1:1 I would not worry too much about vega. The point is to try and earn the spread between the two options. Assume actual vol will be somewhere between the two -- i.e. 20.75%. It doesn't need to be but for the sake of simplicity assume it is. By selling the 111's and delta hedging at 20.75% vol you are replicating the payoff of a properly priced long 111 put. At expiration, the replicated payoff offsets your short and you are left with 0.75 vegas of profit. Same story for the 110s (replicating a properly priced short payoff). Net result -- 1.5 vegas of profit.
To directly answer your question.....there are a couple of ways to realize a profit from vega neutral trading. First of all, remember that vega hedging is needed due to the fact that in practice, volatility is itself a stochastic process. Vol-of-vol creates a whole new risk dimension which must be hedged. One way to make money is the method I outlined before, spreading options against other options and delta hedging to expiration. The crux of it all is that your delta hedges replicate a position in an offsetting option. At expiry, you're left with a stub cashflow (the payoffs all cancel out) which is either your profit or loss. In this case, you were attempting to "arb" IV/RV. The vega hedging was to protect your P&L from shifts in the vol surface.
Another method would be to try and trade vega convexity. Here you are trying to "arb" the other risk dimension. To do this, look for an underlying with a vol surface that is trading flat. If you believe vol is truly stochastic for this underlying, then the OTMs should trade at higher IV's (resulting in a convex vol surface). If not, there is arbitrage. You buy the wings, sell atm, with more wings than atm such that you are vega/delta neutral. Rebalance as needed, and any shift in vol (up or down) should give you positive P&L due to vega convexity (volga) --- i.e. equivalent to gamma scalping in the vega dimension.
Hope this helps.