You can put the WKT text into a text file, and run gdalsrsinfo on it:
gdalsrsinfo test.txt >out.txt
PROJ.4 : '+proj=lcc +lat_1=26.666667 +lat_2=29.333333 +lat_0=28.002808 +lon_0=84 +x_0=500000 +y_0=500000 +a=6377301.243 +b=6356100.230165384 +units=m +no_defs '
The ellipsoid parameters look very much like
Kalianpur 1962, EPSG 4145:
+proj=longlat +a=6377301.243 +b=6356100.230165384 +towgs84=283,682,231,0,0,0,0 +no_defs
Which gives you a 3-parameter transformation to WGS84. But according to the EPSG database the transformation is valid for Pakistan onshore, not India. For India, they suggest EPSG 4146:
+proj=longlat +a=6377299.151 +b=6356098.145120132 +towgs84=295,736,257,0,0,0,0 +no_defs
The ellipsoid is slightly smaller, but the calculated offset is just about 4 centimeters, while the datum shift is given an accuracy of 10 to 25m.
The official NIMA TR8350.2 on the WGS84 Datum lists the following datums on page B.3-2:
Indian based on Everest 1830, a=6377276.345, valid for Bangladesh
Indian based on Everest 1956, a=6377301.243, valid for India and Nepal
Indian based on Everest(Pakistan), a=6377309.613, valid for Pakistan
The flattening is equal for all. Just to confuse, there are other Datums called "Indian" for Thailand, Cambodia and Vietnam, and Malaysia and Sri Lanka use the Everest ellipsoid as well.
Everest 1956 is defined on page A.1-1 with
a=6377301.243 1/f=300.8017 b=6356100.23, just the value you have for Everest 1962. The source notes that the 1956 datum was an adoption of a new yard to meter conversion factor.
See also this article on the conversion of the Everest 1956 datum: http://www.gisdevelopment.net/technology/ip/ma03037pf.htm
The article compares surveyed Everest 1956 coordinates with satellite-measured WGS84 coordinates of 11 major airports, and states offsets between measured and calculated WGS84 coordinates by up to 1 arc second (about 30m). The distortion is arbitrary across the country, so a 7-parameter datum shift will not give better values.
There might have been efforts to get more accurate datum conversions, but the surveying authority is keeping it secret. It might be considered as a case of national security.
See also http://lists.maptools.org/pipermail/proj/2012-November/006472.html and https://github.com/klokantech/epsg.io/issues/49