Add 240v Outlet to Apartment Subpanel
You don't have enough juice for a normal dryer
The problem you have is that while you have space in the panel for the circuits required, your feeder just can't quite provide enough grunt to support a bog-standard electric (condensing) dryer.  In particular, you are looking at 750ft2*3VA/ft2 = 2250VA for lighting, along with 3 1500VA allowances for appliances (microwave, fridge+disposal, kitchen receptacles), which gives you 4313VA of load once you apply the 35% load factor for general loads over 3kVA, or 18A @ 240VAC.  However, that load factor does not apply to the dryer's 5000VA load allowance, and you need to provide 1500VA (load factored, though) for the laundry small appliance circuit as well, which puts you at 9838VA of load, or 41A @ 240VAC, just a hair too large for your 40A feeder.  Oops!
Heat pump dryers to the rescue!
However, there is a way to do what you want without breaking your apartment's feeder, although it does mean buying a not-so-cheap dryer.  You see, in the quest to make a less piggy clothes dryer, some bright sparks hit on the idea of taking the exhaust air from a dryer's drum, running it through what's basically a dehumidifier, and then stuffing it back into the drum.  This arrangement is called a heat pump dryer, and it has the upside that it is exceptionally efficient compared to any other powered clothes-drying means out there, in addition to not requiring a vent, either.
In particular, the Miele TWB1xx series of heat-pump dryers will run perfectly fine on a 120V outlet, pulling roughly 8A from that outlet.  Along with your average apartment-sized washer, with its 8-12A of 120V load, it just barely fits onto a single 120V laundry small appliance branch circuit.  Furthermore, we can skip the clothes dryer allotment in the load calculation as a result of this (allocating 5kVA in a loadcalc for an appliance that cannot ever draw that much power as it plugs into a 120V, 15A branch circuit is simply silly).  As a result, we add one factored 1500VA small appliance allowance to our existing load, giving us 4838VA, or 20A @ 240VAC.
Note that this trickery only works with the Miele dryers -- the Samsung and Blomberg/Beko heat pump dryers both run on 240VAC circuits, and it appears the Samsung requires neutral as well, really constraining one's options when it comes to provisioning power for them.  (Also, Samsung doesn't document how much electricity their dryer actually draws anywhere on their website or in the product specification sheet.)