How are GameStop short sellers going to close their positions when there is an insufficient float?

Short selling without locating shares to borrow is illegal, except for market makers, for this specific reason (along with the unnecessary selling pressure when opening a short position). This is called naked shorting.

The answer is they can't, at once, but they can over time. Maybe they can get GME to create and sell more shares to increase the float, fortunately this is actually the best move for that company as it is the best way they recapitalize.

Let's say 140% of the float was shorted. Obviously the short sellers cannot buy 140% of the float. But they can buy 10%, reducing the amount of available shares by 10%, deliver the shares to people they owe the shares, and that way they amount of available shares goes up again. And they repeat that 14 times.

As I have written before, I have never understood how the short interest could exceed 100% because I always thought that a share could only be loaned out once. The only reason that I could come up with is that they can be loaned multiple times and that seems illogical.So assuming that the latter is true, let me expand on your set up:

+100 A owns the shares and lends them to B

-100 B shorts them, selling them to C

+100 C owns them and lends them to D

-100 D shorts them, selling them to E

+100 E owns the shares (actual physical shares in street name here)

and so on?

If borrowed shares can be re-shorted which in this case is by many institutions, how are they going to cover the positions if let's say GME goes to $1,000? They would be facing a weird situation that the number of float in the market isn't enough to cover their short positions.

In reality, it's highly unlikely that all 140% of the shorters would attempt to cover at once. In addition, if no restrictions were in place, new buyers and shorters would be coming in all the time, making it a fluid situation, kind of a massive shell game with GME shares under each shell.

How are B & D going to recover their positions when the float is only 100 but 200 shares are required? Is the system broken by WSB?

The float will never be just 100 shares (or some low multiple of 100/200). In addition, since there are so many players in the game and a very large number of them are smaller account size Robinhood traders (rather than for example one entity like Porsche short squeezing Volkswagen), there will always be sellers at a higher price.

And lastly, as shorters close their positions during the short squeeze, price soars. When price gets high enough, share owners sell and new shorters come in, driving price down (see GME's price rise to $480 yesterday morning then drop to $120 and subsequent rise to $380 by noon. The cycle is fluid and will repeat but likely diminishing in scale because of trading restrictions being imposed.