Long story short, this archival dataset does not contain precise slice time information. 

Thanks for sharing your BIDS file with me. The crucial bit of information is
  "ManufacturersModelName": "Trio",
  "SoftwareVersions": "syngo_MR_2004A_4VA25A"
This suggests the data comes a Trio running VA generation software from about 2004. To the best of my knowledge, A generation DICOM datasets never included slice timing data. Therefore, it is impossible for dcm2niix to extra this information.
  The sequential generations of Siemens software since about 2003 are:
   VA e.g. Trio running VA25A 
   VB e.g. Trio running VB13
   VC (specific to China) e.g. Trio in China
   VD e.g. Prisma running VD13
   VE e.g. Prisma running VE11
   XA e.g. Vida running XA10 
 Note the final number is the software stepping, with developmental releases being even and deployed releases being odd numbers, so for VB the common versions were VB11, VB13, VB15, VB15 and VB19. In general dcm2niix works well with VB through VE. The XA is a dramatically different DICOM format, and Siemens still seems to be tweaking the data included in it. At the moment, the XA format omits many of the useful fields we are used to for research data