Roughly, I find that the net melting of global sea ice is adding about 10% to the enhanced greenhouse effect.
Some numbers. In a comment, S. Davis wrote:
As you know, there is a distinct lack of ice this year, and there is even less than in 2007. However, like 2007, there is an abundance of ice in Antarctica. In fact, we are above the 1979-2000 satellite average by about 1 million square km. And just last week a paper was released that showed that the Antarctic has been gaining ice mass since 2003, and not losing ice since 1992. As you know, the Antarctic contains about 90% of the ice on this planet, while the Arctic contains about .01% of the global total, that is one ten thousandth of the total. The Antarctic contains about 90% of the ice on this planet. Any small gain in the Antarctic dwarfs any loss in the Arctic. For example, a 1% increase in the Antarctic is almost 300,000 cubic km., while the Arctic only contains 24,000 cubic km. in winter!In fact, Arctic sea ice is melting faster than Antarctic sea ice is increasing, with the Antarctic currently having about 53% of total sea ice. In the last 10 years, Arctic SIE has decreased by 7.3% relative to the 10 years before, while Antarctic SIE has increased by 2.3%.
Total SIE has decreased by 2.5%, and the trend of total SIE extent since the satellite era -38 Kkm2/yr. (It's -47 Kkm2/yr over the last decade.)
This net melting changes the Earth's albedo, and it absorbs more energy. If S is the solar radiation incident on a piece of sea ice of area A, and it melts, the net energy absorbed is
ΔE = (αice - αsea)SA
where α is the albedo. αice ≈ 0.5 and αsea ≈ 0.1.
David Randall's book has a chart (pg 33) showing that the average solar constant at 80°N during the time when the Sun is above the horizon is 250 W/m2, and the Antarctic receives very nearly the same. Let's divide S by 2, since the Sun is up only half the year above [below] the Arctic [Antarctic] circle.
Using these numbers, I find the energy absorbed from this melting ice is an additional 2 trillion watts per year. Averaged over the Earth's entire surface it comes to an additional 0.004 W/m2 of energy each year, over the entire globe
(This additional heat isn't strictly a forcing, since it takes place regionally and not globally, but this gives an idea of its magnitude. Also, note that I have not assumed any melting of Antarctic land ice.)
Global anthropogenic radiative forcing from greenhouse gases is now about 2.8 W/m2, increasing about 0.03 W/m2 each year. (The net forcing is about half this, due to the cooling effect of aerosols.)
So just the melting sea ice is adding roughly 10% to the manmade greenhouse effect -- not an insubstantial amount.