The 4210U CPU has a TDP of only 15w and a max temp of 100c. It should be very easy to build a fanless solution with that and a NUC style case. Many processors with a 15w TDP use fanless heat sinks so it should be possible to either use a third party low noise option or slow the stock fan down with a speed controller. (I've actually unplugged or removed the fan on low powered CPU/GPU heat sinks with no issues but temps must be monitored.)
Another option may be something like the GIGABYTE GB-BXi3-4010 or GIGABYTE GB-BXi5-4200. They have a fan but it shouldn't be difficult to fix if you don't mind replacing the heat sink. I've used third party low profile "silent" heat sinks in HTPCs with good results. The GIGABYTE BRIX line tends to be a little richer in features than Intel NUC line.
The HDPlex cases are very nice but keep in mind that they cost as much as some low end mini-PCs.