Ligase is the enzyme that will fix this lagging strand. When DNA is replicated for cell division, only one of the strands can be replicated in one continuous strand. The other half of the DNA is replicated in short spurts. This short spurt results in the kind of thing seen in the picture. The ligase binds the short, lagging pieces. The ligase will also fix any missing pieces in a DNA strand so long as it has a template to work with. However, there’s also a kind of ligase that’ll fix a break occurring in both strands of DNA at the same place.
In non-technical terms, a ligase will make the same kind of phosphorous base that is used to make the backbone of the DNA chain. In short, that’s how it binds the DNA.