The mitotic apparatus is a transient cell structure that is used for cell division. This includes the splitting and reforming of DNA strands into, firstly two halves, and then into two complete strands, which are polarised into two ends of the cell on spindles of tanulin, ready for the last part, splitting and reforming the phosolipid bi-layer of the parent cell's wall to form two separate cells. This is a very useful device for research purposes in DNA production, DNA amino acid isolation and general splicing. Here is a suggestion, although the three part structure of the mitotic apparatus has only relatively recently been located in a studiable fashion, and there are still chemical components that have yet to be identified.
After finding out which chemical elements are actually in the molecular structure of every part of the mitotic apparatus (tanulin in the spindles, calcium for initiation and inhibition, zinc likewise, also what "motivates" kinesin to move, and what the rest of the mechanism is made of), which involves breaking samples down, trying to create a similarly acting device through nanotechnology could prove to be enlightening whether the goal is reached or not.
Firstly, trying to chemically recreate the device through nanotechnology will inevitably show the ease in some things and, more interestingly, the difficulty in other parts. This will lead to a greater understanding of both why and how certain features of biology occur on a chemical level. This is great for medical research, as well as general knowledge in the fields of biochemistry and genetics. There will be mysteries and surprises, but that is where the discoveries and breakthroughs lie.
If the rather complex mitotic apparatus is successfully artificially built, the uses for it are boundless. You could post any DNA strand through for splitting, breaking down and doing research, as well as combining and building. It could be used for the "unwinding" of DNA (taking apart), if the full stages of the mitotis is cut short (maybe by having no cell wall in some cases). This is great for gene isolation, and research by deduction and/or mutation. It could be used, in future medical sciences, to splice a gene, or gene sequences, in or out of the daughter cells. Working with what is already there, it can be used to turn genes on and off. It can, naturally, be used to "put things together", as well as pull them apart. It is better to work WITH how the biology is designed to (with prods in the wanted direction), then against it (inhibition, chemical blocking, physical removal). For example, initiating a cancer specific cell death is both what the body is trying to do, and has a millennia designed mechanism already in place (cell suicide for the health of the greater organism). However, trying to block cancer cell reproduction is swimming against a tide of evolutionary built mechanisms (cell recreation and growth with the chemical signals normally reserved or interpreted as such by embryos, foetuses and infants; have you ever noticed that many cancer cells behave like the newly conceived of any animal in their explosive growth?), although working on it from both ends will probably result in a more thorough treatment.
Of course any ensuing medical industries would have to be closely monitored, so the technologies are used voluntarily by patients wishing to prevent the suffering in genetically inherited diseases and conditions that are life threatening or high maintenance, such as cancers, congenital blindness, cerebral palsy, dangerous allergies, severe growth disorders etc., as opposed to "designer" babies with just the right shade of eye colour or ability.